{"id":1177,"date":"2026-09-14T13:34:56","date_gmt":"2026-09-14T13:34:56","guid":{"rendered":"https:\/\/otus.ru\/journal\/?p=1177"},"modified":"2026-09-25T15:52:49","modified_gmt":"2026-09-25T15:52:49","slug":"kak-vozvesti-matricu-v-stepen","status":"publish","type":"post","link":"https:\/\/otus.ru\/journal\/kak-vozvesti-matricu-v-stepen\/","title":{"rendered":"\u041a\u0430\u043a \u0432\u043e\u0437\u0432\u0435\u0441\u0442\u0438 \u043c\u0430\u0442\u0440\u0438\u0446\u0443 \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c: \u043f\u043e\u0448\u0430\u0433\u043e\u0432\u044b\u0439 \u0440\u0430\u0437\u0431\u043e\u0440 \u043d\u0430 \u043f\u0440\u0438\u043c\u0435\u0440\u0435 2&#215;2"},"content":{"rendered":"<p><strong>\u0412\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043c\u0430\u0442\u0440\u0438\u0446\u044b \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c<\/strong> &#8212; \u044d\u0442\u043e \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043d\u043e\u0439 \u043c\u0430\u0442\u0440\u0438\u0446\u044b \u043d\u0430 \u0441\u0430\u043c\u0443 \u0441\u0435\u0431\u044f n \u0440\u0430\u0437, \u0433\u0434\u0435 n &#8212; \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439 \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c. \u0417\u0430\u043f\u0438\u0441\u044c A^n \u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442 A * A * &#8230; * A (n \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u0439), \u0438 \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u0437\u0434\u0435\u0441\u044c \u043c\u0430\u0442\u0440\u0438\u0447\u043d\u043e\u0435 (\u0441\u0442\u0440\u043e\u043a\u0430 \u043d\u0430 \u0441\u0442\u043e\u043b\u0431\u0435\u0446), \u0430 \u043d\u0435 \u043f\u043e\u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u043e\u0435. \u041d\u0438\u0436\u0435 \u044f \u043f\u043e \u0448\u0430\u0433\u0430\u043c \u043f\u043e\u0441\u0447\u0438\u0442\u0430\u044e A^2 \u0438 A^3 \u0434\u043b\u044f \u043a\u043e\u043d\u043a\u0440\u0435\u0442\u043d\u043e\u0439 \u043c\u0430\u0442\u0440\u0438\u0446\u044b 2&#215;2, \u0440\u0443\u043a\u0430\u043c\u0438 \u0438 \u0441 \u0447\u0438\u0441\u043b\u0430\u043c\u0438, \u043f\u043e\u0442\u043e\u043c \u043f\u0440\u043e\u0432\u0435\u0440\u044e \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u043a\u043e\u0434\u043e\u043c \u043d\u0430 numpy \u0438 \u043f\u043e\u043a\u0430\u0436\u0443, \u0447\u0435\u043c \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u043c\u0430\u0442\u0440\u0438\u0446\u044b \u043e\u0442\u043b\u0438\u0447\u0430\u0435\u0442\u0441\u044f \u043e\u0442 \u043f\u043e\u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u043e\u0433\u043e \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f.<\/p>\n<div class=\"oj-toc\" style=\"background:#f6f6f6;border-radius:8px;padding:14px 20px;margin:20px 0\">\n<p><strong>\u0421\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435<\/strong><\/p>\n<ol>\n<li><a href=\"#s1\">\u041a\u043e\u0433\u0434\u0430 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f \u0432\u043e\u043e\u0431\u0449\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0430<\/a><\/li>\n<li><a href=\"#s2\">\u0412\u043e\u0437\u044c\u043c\u0435\u043c \u043a\u043e\u043d\u043a\u0440\u0435\u0442\u043d\u0443\u044e \u043c\u0430\u0442\u0440\u0438\u0446\u0443<\/a><\/li>\n<li><a href=\"#s3\">\u0428\u0430\u0433 1. \u0421\u0447\u0438\u0442\u0430\u0435\u043c A^2 (\u043a\u0432\u0430\u0434\u0440\u0430\u0442 \u043c\u0430\u0442\u0440\u0438\u0446\u044b) \u0440\u0443\u043a\u0430\u043c\u0438<\/a><\/li>\n<li><a href=\"#s4\">\u0428\u0430\u0433 2. \u0421\u0447\u0438\u0442\u0430\u0435\u043c A^3 = A^2 * A<\/a><\/li>\n<li><a href=\"#s5\">\u0421\u0442\u0435\u043f\u0435\u043d\u044c \u043c\u0430\u0442\u0440\u0438\u0446\u044b &#8212; \u044d\u0442\u043e \u043d\u0435 \u043f\u043e\u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u043e\u0435 \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435<\/a><\/li>\n<li><a href=\"#s6\">\u041f\u0440\u043e\u0432\u0435\u0440\u043a\u0430 \u043a\u043e\u0434\u043e\u043c \u043d\u0430 numpy<\/a><\/li>\n<li><a href=\"#s7\">\u041a\u043e\u0440\u043e\u0442\u043a\u043e \u043f\u0440\u043e \u0431\u043e\u043b\u044c\u0448\u0438\u0435 \u0441\u0442\u0435\u043f\u0435\u043d\u0438<\/a><\/li>\n<li><a href=\"#s8\">\u0412\u044b\u0432\u043e\u0434\u044b<\/a><\/li>\n<li><a href=\"#s9\">\u0413\u0434\u0435 \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f \/ \u0441\u0432\u044f\u0437\u044c \u0441 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u043e\u0439<\/a><\/li>\n<li><a href=\"#s10\">FAQ<\/a><\/li>\n<\/ol>\n<\/div>\n<h2 id=\"s1\">\u041a\u043e\u0433\u0434\u0430 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f \u0432\u043e\u043e\u0431\u0449\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0430<\/h2>\n<p>\u0423 \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u0435\u0441\u0442\u044c \u0436\u0435\u0441\u0442\u043a\u0438\u0435 \u0433\u0440\u0430\u043d\u0438\u0446\u044b \u043f\u0440\u0438\u043c\u0435\u043d\u0438\u043c\u043e\u0441\u0442\u0438 &#8212; \u0431\u0435\u0437 \u043d\u0438\u0445 \u0437\u0430\u043f\u0438\u0441\u044c A^n \u043d\u0435 \u0438\u043c\u0435\u0435\u0442 \u0441\u043c\u044b\u0441\u043b\u0430.<\/p>\n<ul>\n<li><strong>\u0422\u043e\u043b\u044c\u043a\u043e \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043d\u0430\u044f \u043c\u0430\u0442\u0440\u0438\u0446\u0430.<\/strong> \u0427\u0438\u0441\u043b\u043e \u0441\u0442\u0440\u043e\u043a \u0440\u0430\u0432\u043d\u043e \u0447\u0438\u0441\u043b\u0443 \u0441\u0442\u043e\u043b\u0431\u0446\u043e\u0432 (2&#215;2, 3&#215;3 \u0438 \u0442\u0430\u043a \u0434\u0430\u043b\u0435\u0435). \u0423\u043c\u043d\u043e\u0436\u0430\u0442\u044c \u043c\u0430\u0442\u0440\u0438\u0446\u0443 \u043d\u0430 \u0441\u0430\u043c\u0443 \u0441\u0435\u0431\u044f \u043c\u043e\u0436\u043d\u043e \u043b\u0438\u0448\u044c \u0442\u043e\u0433\u0434\u0430, \u043a\u043e\u0433\u0434\u0430 \u0447\u0438\u0441\u043b\u043e \u0441\u0442\u043e\u043b\u0431\u0446\u043e\u0432 \u043f\u0435\u0440\u0432\u043e\u0433\u043e \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044f \u0441\u043e\u0432\u043f\u0430\u0434\u0430\u0435\u0442 \u0441 \u0447\u0438\u0441\u043b\u043e\u043c \u0441\u0442\u0440\u043e\u043a \u0432\u0442\u043e\u0440\u043e\u0433\u043e, \u0430 \u044d\u0442\u043e \u0432\u0435\u0440\u043d\u043e \u0442\u043e\u043b\u044c\u043a\u043e \u0434\u043b\u044f \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043d\u043e\u0439.<\/li>\n<li><strong>\u041f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c &#8212; \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e<\/strong> (1, 2, 3, &#8230;). \u0414\u0440\u043e\u0431\u043d\u044b\u0435 \u0438 \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0435 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 &#8212; \u044d\u0442\u043e \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u0430\u044f \u0442\u0435\u043c\u0430 (\u0434\u043b\u044f \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0445 \u043d\u0443\u0436\u043d\u0430 \u043e\u0431\u0440\u0430\u0442\u043d\u0430\u044f \u043c\u0430\u0442\u0440\u0438\u0446\u0430, \u0438 \u043e\u043d\u0430 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u043d\u0435 \u0432\u0441\u0435\u0433\u0434\u0430), \u0432 \u044d\u0442\u043e\u0439 \u0441\u0442\u0430\u0442\u044c\u0435 \u044f \u0438\u0445 \u043d\u0435 \u0442\u0440\u043e\u0433\u0430\u044e.<\/li>\n<li><strong>\u0413\u0440\u0430\u043d\u0438\u0447\u043d\u044b\u0435 \u0441\u043b\u0443\u0447\u0430\u0438:<\/strong> A^1 = A (\u0441\u0430\u043c\u0430 \u043c\u0430\u0442\u0440\u0438\u0446\u0430), A^0 = E (\u0435\u0434\u0438\u043d\u0438\u0447\u043d\u0430\u044f \u043c\u0430\u0442\u0440\u0438\u0446\u0430 \u0442\u043e\u0433\u043e \u0436\u0435 \u0440\u0430\u0437\u043c\u0435\u0440\u0430, \u0443 \u043d\u0435\u0435 \u0435\u0434\u0438\u043d\u0438\u0446\u044b \u043d\u0430 \u0433\u043b\u0430\u0432\u043d\u043e\u0439 \u0434\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u0438 \u0438 \u043d\u0443\u043b\u0438 \u0432\u043d\u0435 \u0435\u0435). \u042d\u0442\u043e \u0441\u043e\u0433\u043b\u0430\u0448\u0435\u043d\u0438\u0435, \u0430\u043d\u0430\u043b\u043e\u0433 \u0442\u043e\u0433\u043e, \u0447\u0442\u043e \u043b\u044e\u0431\u043e\u0435 \u0447\u0438\u0441\u043b\u043e \u0432 \u043d\u0443\u043b\u0435\u0432\u043e\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 \u0440\u0430\u0432\u043d\u043e 1.<\/li>\n<\/ul>\n<p>\u0415\u0434\u0438\u043d\u0438\u0447\u043d\u0430\u044f \u043c\u0430\u0442\u0440\u0438\u0446\u0430 2&#215;2 \u0432\u044b\u0433\u043b\u044f\u0434\u0438\u0442 \u0442\u0430\u043a:<\/p>\n<table>\n<thead>\n<tr>\n<th><\/th>\n<th><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 id=\"s2\">\u0412\u043e\u0437\u044c\u043c\u0435\u043c \u043a\u043e\u043d\u043a\u0440\u0435\u0442\u043d\u0443\u044e \u043c\u0430\u0442\u0440\u0438\u0446\u0443<\/h2>\n<p>\u0414\u0430\u043b\u044c\u0448\u0435 \u0432\u0435\u0441\u044c \u0440\u0430\u0437\u0431\u043e\u0440 \u0438\u0434\u0435\u0442 \u043d\u0430 \u043e\u0434\u043d\u043e\u0439 \u043c\u0430\u0442\u0440\u0438\u0446\u0435 A \u0440\u0430\u0437\u043c\u0435\u0440\u0430 2&#215;2 &#8212; \u043e\u043d\u0430 \u0431\u0443\u0434\u0435\u0442 \u0441\u043a\u0432\u043e\u0437\u043d\u043e\u0439 \u0447\u0435\u0440\u0435\u0437 \u0432\u0441\u044e \u0441\u0442\u0430\u0442\u044c\u044e:<\/p>\n<table>\n<thead>\n<tr>\n<th><\/th>\n<th><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u0422\u043e \u0435\u0441\u0442\u044c A = [[1, 2], [3, 4]]: \u0432 \u043f\u0435\u0440\u0432\u043e\u0439 \u0441\u0442\u0440\u043e\u043a\u0435 1 \u0438 2, \u0432\u043e \u0432\u0442\u043e\u0440\u043e\u0439 3 \u0438 4.<\/p>\n<h2 id=\"s3\">\u0428\u0430\u0433 1. \u0421\u0447\u0438\u0442\u0430\u0435\u043c A^2 (\u043a\u0432\u0430\u0434\u0440\u0430\u0442 \u043c\u0430\u0442\u0440\u0438\u0446\u044b) \u0440\u0443\u043a\u0430\u043c\u0438<\/h2>\n<p>A^2 = A * A. \u041f\u0440\u0430\u0432\u0438\u043b\u043e \u043c\u0430\u0442\u0440\u0438\u0447\u043d\u043e\u0433\u043e \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u044f: \u044d\u043b\u0435\u043c\u0435\u043d\u0442 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u0430 \u0432 \u043f\u043e\u0437\u0438\u0446\u0438\u0438 (\u0441\u0442\u0440\u043e\u043a\u0430 i, \u0441\u0442\u043e\u043b\u0431\u0435\u0446 j) &#8212; \u044d\u0442\u043e \u0441\u0443\u043c\u043c\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0439 \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043e\u0432 i-\u0439 \u0441\u0442\u0440\u043e\u043a\u0438 \u043f\u0435\u0440\u0432\u043e\u0439 \u043c\u0430\u0442\u0440\u0438\u0446\u044b \u043d\u0430 \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u044b j-\u0433\u043e \u0441\u0442\u043e\u043b\u0431\u0446\u0430 \u0432\u0442\u043e\u0440\u043e\u0439 \u043c\u0430\u0442\u0440\u0438\u0446\u044b. \u041f\u0440\u043e\u0449\u0435 \u0433\u043e\u0432\u043e\u0440\u044f, \u00ab\u0441\u0442\u0440\u043e\u043a\u0430 \u043d\u0430 \u0441\u0442\u043e\u043b\u0431\u0435\u0446\u00bb.<\/p>\n<p>\u0423 \u043d\u0430\u0441 \u043e\u0431\u0435 \u043c\u0430\u0442\u0440\u0438\u0446\u044b \u043e\u0434\u0438\u043d\u0430\u043a\u043e\u0432\u044b\u0435 (A * A), \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0431\u0435\u0440\u0435\u043c \u0441\u0442\u0440\u043e\u043a\u0438 \u043b\u0435\u0432\u043e\u0439 A \u0438 \u0441\u0442\u043e\u043b\u0431\u0446\u044b \u043f\u0440\u0430\u0432\u043e\u0439 A. \u0421\u0447\u0438\u0442\u0430\u0435\u043c \u0432\u0441\u0435 \u0447\u0435\u0442\u044b\u0440\u0435 \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u0430.<\/p>\n<ul>\n<li>\u041f\u043e\u0437\u0438\u0446\u0438\u044f (1, 1): 1-\u044f \u0441\u0442\u0440\u043e\u043a\u0430 (1, 2) \u043d\u0430 1-\u0439 \u0441\u0442\u043e\u043b\u0431\u0435\u0446 (1, 3): <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">1*1 + 2*3 = 1 + 6 = 7<\/code>.<\/li>\n<li>\u041f\u043e\u0437\u0438\u0446\u0438\u044f (1, 2): 1-\u044f \u0441\u0442\u0440\u043e\u043a\u0430 (1, 2) \u043d\u0430 2-\u0439 \u0441\u0442\u043e\u043b\u0431\u0435\u0446 (2, 4): <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">1*2 + 2*4 = 2 + 8 = 10<\/code>.<\/li>\n<li>\u041f\u043e\u0437\u0438\u0446\u0438\u044f (2, 1): 2-\u044f \u0441\u0442\u0440\u043e\u043a\u0430 (3, 4) \u043d\u0430 1-\u0439 \u0441\u0442\u043e\u043b\u0431\u0435\u0446 (1, 3): <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">3*1 + 4*3 = 3 + 12 = 15<\/code>.<\/li>\n<li>\u041f\u043e\u0437\u0438\u0446\u0438\u044f (2, 2): 2-\u044f \u0441\u0442\u0440\u043e\u043a\u0430 (3, 4) \u043d\u0430 2-\u0439 \u0441\u0442\u043e\u043b\u0431\u0435\u0446 (2, 4): <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">3*2 + 4*4 = 6 + 16 = 22<\/code>.<\/li>\n<\/ul>\n<p>\u0421\u043e\u0431\u0438\u0440\u0430\u0435\u043c \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442:<\/p>\n<table>\n<thead>\n<tr>\n<th><\/th>\n<th><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>7<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>15<\/td>\n<td>22<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u0418\u0442\u043e\u0433\u043e A^2 = [[7, 10], [15, 22]].<\/p>\n<h2 id=\"s4\">\u0428\u0430\u0433 2. \u0421\u0447\u0438\u0442\u0430\u0435\u043c A^3 = A^2 * A<\/h2>\n<p>\u0414\u043b\u044f \u0442\u0440\u0435\u0442\u044c\u0435\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 \u043d\u0435 \u043d\u0443\u0436\u043d\u043e \u043f\u0435\u0440\u0435\u043c\u043d\u043e\u0436\u0430\u0442\u044c \u0442\u0440\u0438 \u043c\u0430\u0442\u0440\u0438\u0446\u044b \u0441\u0440\u0430\u0437\u0443 &#8212; \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u0432\u0437\u044f\u0442\u044c \u0443\u0436\u0435 \u043f\u043e\u0441\u0447\u0438\u0442\u0430\u043d\u043d\u0443\u044e A^2 \u0438 \u0443\u043c\u043d\u043e\u0436\u0438\u0442\u044c \u0435\u0435 \u043d\u0430 A. \u042d\u0442\u043e \u043a\u043b\u044e\u0447\u0435\u0432\u0430\u044f \u0438\u0434\u0435\u044f: A^3 = A^2 * A, A^4 = A^3 * A \u0438 \u0442\u0430\u043a \u0434\u0430\u043b\u0435\u0435, \u043a\u0430\u0436\u0434\u044b\u0439 \u0440\u0430\u0437 \u0434\u043e\u043c\u043d\u043e\u0436\u0430\u0435\u043c \u043f\u0440\u0435\u0434\u044b\u0434\u0443\u0449\u0438\u0439 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u043d\u0430 A.<\/p>\n<p>\u0411\u0435\u0440\u0435\u043c \u0441\u0442\u0440\u043e\u043a\u0438 A^2 = [[7, 10], [15, 22]] \u0438 \u0441\u0442\u043e\u043b\u0431\u0446\u044b A = [[1, 2], [3, 4]].<\/p>\n<ul>\n<li>\u041f\u043e\u0437\u0438\u0446\u0438\u044f (1, 1): (7, 10) \u043d\u0430 (1, 3): <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">7*1 + 10*3 = 7 + 30 = 37<\/code>.<\/li>\n<li>\u041f\u043e\u0437\u0438\u0446\u0438\u044f (1, 2): (7, 10) \u043d\u0430 (2, 4): <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">7*2 + 10*4 = 14 + 40 = 54<\/code>.<\/li>\n<li>\u041f\u043e\u0437\u0438\u0446\u0438\u044f (2, 1): (15, 22) \u043d\u0430 (1, 3): <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">15*1 + 22*3 = 15 + 66 = 81<\/code>.<\/li>\n<li>\u041f\u043e\u0437\u0438\u0446\u0438\u044f (2, 2): (15, 22) \u043d\u0430 (2, 4): <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">15*2 + 22*4 = 30 + 88 = 118<\/code>.<\/li>\n<\/ul>\n<p>\u0420\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442:<\/p>\n<table>\n<thead>\n<tr>\n<th><\/th>\n<th><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>37<\/td>\n<td>54<\/td>\n<\/tr>\n<tr>\n<td>81<\/td>\n<td>118<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u0418\u0442\u043e\u0433\u043e A^3 = [[37, 54], [81, 118]].<\/p>\n<p>\u041f\u043e\u0440\u044f\u0434\u043e\u043a \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u0439 \u0432 \u043e\u0431\u0449\u0435\u043c \u0441\u043b\u0443\u0447\u0430\u0435 \u0432\u0430\u0436\u0435\u043d (A * B \u043d\u0435 \u0432\u0441\u0435\u0433\u0434\u0430 \u0440\u0430\u0432\u043d\u043e B * A), \u043d\u043e \u043a\u043e\u043d\u043a\u0440\u0435\u0442\u043d\u043e \u043f\u0440\u0438 \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0438 \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u043c\u044b \u0443\u043c\u043d\u043e\u0436\u0430\u0435\u043c \u043c\u0430\u0442\u0440\u0438\u0446\u0443 \u043d\u0430 \u0441\u0430\u043c\u0443 \u0441\u0435\u0431\u044f, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 A^2 * A \u0438 A * A^2 \u0434\u0430\u0434\u0443\u0442 \u043e\u0434\u043d\u043e \u0438 \u0442\u043e \u0436\u0435.<\/p>\n<h2 id=\"s5\">\u0421\u0442\u0435\u043f\u0435\u043d\u044c \u043c\u0430\u0442\u0440\u0438\u0446\u044b &#8212; \u044d\u0442\u043e \u043d\u0435 \u043f\u043e\u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u043e\u0435 \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435<\/h2>\n<p>\u0417\u0434\u0435\u0441\u044c \u0447\u0430\u0449\u0435 \u0432\u0441\u0435\u0433\u043e \u043e\u0448\u0438\u0431\u0430\u044e\u0442\u0441\u044f. \u0412\u043e\u0437\u0432\u0435\u0441\u0442\u0438 \u043c\u0430\u0442\u0440\u0438\u0446\u0443 \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u0438 \u0432\u043e\u0437\u0432\u0435\u0441\u0442\u0438 \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u043a\u0430\u0436\u0434\u044b\u0439 \u0435\u0435 \u044d\u043b\u0435\u043c\u0435\u043d\u0442 \u043f\u043e \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u0438 &#8212; \u0434\u0432\u0435 \u0440\u0430\u0437\u043d\u044b\u0435 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438 \u0441 \u0440\u0430\u0437\u043d\u044b\u043c\u0438 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u0430\u043c\u0438.<\/p>\n<p>\u0415\u0441\u043b\u0438 \u0431\u044b \u043c\u044b \u043f\u0440\u043e\u0441\u0442\u043e \u0432\u043e\u0437\u0432\u0435\u043b\u0438 \u043a\u0430\u0436\u0434\u044b\u0439 \u044d\u043b\u0435\u043c\u0435\u043d\u0442 A \u0432 \u043a\u0432\u0430\u0434\u0440\u0430\u0442 (\u043f\u043e\u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u043e), \u043f\u043e\u043b\u0443\u0447\u0438\u043b\u0438 \u0431\u044b [[1, 4], [9, 16]]: <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">1^2=1<\/code>, <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">2^2=4<\/code>, <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">3^2=9<\/code>, <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">4^2=16<\/code>. \u0421\u0440\u0430\u0432\u043d\u0438\u0442\u0435 \u0441 \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0438\u043c A^2:<\/p>\n<table>\n<thead>\n<tr>\n<th>\u041e\u043f\u0435\u0440\u0430\u0446\u0438\u044f<\/th>\n<th>\u0427\u0442\u043e \u0434\u0435\u043b\u0430\u0435\u043c<\/th>\n<th>\u0420\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u0434\u043b\u044f \u043d\u0430\u0448\u0435\u0439 A<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u0421\u0442\u0435\u043f\u0435\u043d\u044c \u043c\u0430\u0442\u0440\u0438\u0446\u044b (A^2)<\/td>\n<td>\u041c\u0430\u0442\u0440\u0438\u0447\u043d\u043e\u0435 \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 A \u043d\u0430 A (\u0441\u0442\u0440\u043e\u043a\u0430 \u043d\u0430 \u0441\u0442\u043e\u043b\u0431\u0435\u0446)<\/td>\n<td>[[7, 10], [15, 22]]<\/td>\n<\/tr>\n<tr>\n<td>\u041f\u043e\u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u043e\u0435 \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435<\/td>\n<td>\u041a\u0430\u0436\u0434\u044b\u0439 \u044d\u043b\u0435\u043c\u0435\u043d\u0442 \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u043e \u0432 \u043a\u0432\u0430\u0434\u0440\u0430\u0442<\/td>\n<td>[[1, 4], [9, 16]]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u0427\u0438\u0441\u043b\u0430 \u043d\u0435 \u0441\u043e\u0432\u043f\u0430\u0434\u0430\u044e\u0442 \u043d\u0438 \u0432 \u043e\u0434\u043d\u043e\u0439 \u043a\u043b\u0435\u0442\u043a\u0435. \u041a\u043e\u0433\u0434\u0430 \u0433\u043e\u0432\u043e\u0440\u044f\u0442 \u00ab\u0432\u043e\u0437\u0432\u0435\u0441\u0442\u0438 \u043c\u0430\u0442\u0440\u0438\u0446\u0443 \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c\u00bb, \u043f\u043e \u0443\u043c\u043e\u043b\u0447\u0430\u043d\u0438\u044e \u0438\u043c\u0435\u044e\u0442 \u0432 \u0432\u0438\u0434\u0443 \u0438\u043c\u0435\u043d\u043d\u043e \u043c\u0430\u0442\u0440\u0438\u0447\u043d\u043e\u0435 \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435, \u0442\u043e \u0435\u0441\u0442\u044c \u043f\u0435\u0440\u0432\u0443\u044e \u0441\u0442\u0440\u043e\u043a\u0443 \u0442\u0430\u0431\u043b\u0438\u0446\u044b. \u041f\u043e\u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u0430\u044f \u0432\u0435\u0440\u0441\u0438\u044f &#8212; \u044d\u0442\u043e \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u0430\u044f \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f, \u0432 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0437\u0430\u043f\u0438\u0441\u0438 \u0435\u0435 \u043e\u0431\u044b\u0447\u043d\u043e \u043e\u0431\u043e\u0437\u043d\u0430\u0447\u0430\u044e\u0442 \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u043e (\u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, A \u0441 \u043a\u0440\u0443\u0436\u043a\u043e\u043c-\u0441\u0442\u0435\u043f\u0435\u043d\u044c\u044e), \u0430 \u0432 \u043a\u043e\u0434\u0435 numpy \u043e\u043d\u0430 \u0434\u0435\u043b\u0430\u0435\u0442\u0441\u044f \u043e\u043f\u0435\u0440\u0430\u0442\u043e\u0440\u043e\u043c <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">**<\/code>, \u043e \u0447\u0435\u043c \u043d\u0438\u0436\u0435.<\/p>\n<h2 id=\"s6\">\u041f\u0440\u043e\u0432\u0435\u0440\u043a\u0430 \u043a\u043e\u0434\u043e\u043c \u043d\u0430 numpy<\/h2>\n<p>\u0420\u0443\u0447\u043d\u043e\u0439 \u0441\u0447\u0435\u0442 \u0443\u0434\u043e\u0431\u043d\u043e \u043f\u0440\u043e\u0432\u0435\u0440\u044f\u0442\u044c \u0438 \u0430\u0432\u0442\u043e\u043c\u0430\u0442\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u0442\u044c. \u0412 Python \u0434\u043b\u044f \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0435\u0433\u043e \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u043c\u0430\u0442\u0440\u0438\u0446\u044b \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u0435\u0441\u0442\u044c \u0433\u043e\u0442\u043e\u0432\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">numpy.linalg.matrix_power<\/code> &#8212; \u0435\u0435 \u0438 \u0441\u0442\u043e\u0438\u0442 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c, \u0430 \u043d\u0435 \u043f\u0438\u0441\u0430\u0442\u044c \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u0432\u0440\u0443\u0447\u043d\u0443\u044e.<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">import<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">numpy<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #C678DD\">as<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">np<\/span>\n\n<span style=\"color: #E06C75\">A<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">array<\/span><span style=\"color: #ABB2BF\">([[<\/span><span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">2<\/span><span style=\"color: #ABB2BF\">],<\/span>\n              <span style=\"color: #ABB2BF\">[<\/span><span style=\"color: #D19A66\">3<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">4<\/span><span style=\"color: #ABB2BF\">]])<\/span>\n\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">linalg<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">matrix_power<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">A<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">2<\/span><span style=\"color: #ABB2BF\">))<\/span>\n<span style=\"color: #7F848E\"># \u0412\u044b\u0432\u043e\u0434:<\/span>\n<span style=\"color: #7F848E\"># [[ 7 10]<\/span>\n<span style=\"color: #7F848E\">#  [15 22]]<\/span>\n\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">linalg<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">matrix_power<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">A<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">3<\/span><span style=\"color: #ABB2BF\">))<\/span>\n<span style=\"color: #7F848E\"># \u0412\u044b\u0432\u043e\u0434:<\/span>\n<span style=\"color: #7F848E\"># [[ 37  54]<\/span>\n<span style=\"color: #7F848E\">#  [ 81 118]]<\/span>\n\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">linalg<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">matrix_power<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">A<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">))<\/span>\n<span style=\"color: #7F848E\"># \u0412\u044b\u0432\u043e\u0434 (\u0435\u0434\u0438\u043d\u0438\u0447\u043d\u0430\u044f \u043c\u0430\u0442\u0440\u0438\u0446\u0430 E):<\/span>\n<span style=\"color: #7F848E\"># [[1 0]<\/span>\n<span style=\"color: #7F848E\">#  [0 1]]<\/span><\/code><\/pre>\n<p>\u0427\u0438\u0441\u043b\u0430 \u0441\u043e\u0432\u043f\u0430\u043b\u0438 \u0441 \u0442\u0435\u043c, \u0447\u0442\u043e \u044f \u043f\u043e\u0441\u0447\u0438\u0442\u0430\u043b \u0440\u0443\u043a\u0430\u043c\u0438, &#8212; \u0437\u043d\u0430\u0447\u0438\u0442, \u0430\u0440\u0438\u0444\u043c\u0435\u0442\u0438\u043a\u0430 \u0432\u0435\u0440\u043d\u0430. \u041e\u0431\u0440\u0430\u0442\u0438\u0442\u0435 \u0432\u043d\u0438\u043c\u0430\u043d\u0438\u0435 \u043d\u0430 \u0433\u0440\u0430\u043d\u0438\u0446\u0443 \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442\u0430: <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">matrix_power<\/code> \u0434\u0435\u043b\u0430\u0435\u0442 \u0438\u043c\u0435\u043d\u043d\u043e \u043c\u0430\u0442\u0440\u0438\u0447\u043d\u043e\u0435 \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435. \u0410 \u0432\u043e\u0442 \u043e\u043f\u0435\u0440\u0430\u0442\u043e\u0440 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">**<\/code> \u0432 numpy \u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442 \u043f\u043e\u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u043e \u0438 \u0434\u0430\u0441\u0442 \u0441\u043e\u0432\u0441\u0435\u043c \u0434\u0440\u0443\u0433\u043e\u0435:<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">A<\/span> <span style=\"color: #56B6C2\">**<\/span> <span style=\"color: #D19A66\">2<\/span><span style=\"color: #ABB2BF\">)<\/span>\n<span style=\"color: #7F848E\"># \u0412\u044b\u0432\u043e\u0434 (\u043f\u043e\u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u043e, \u041d\u0415 \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u043c\u0430\u0442\u0440\u0438\u0446\u044b):<\/span>\n<span style=\"color: #7F848E\"># [[ 1  4]<\/span>\n<span style=\"color: #7F848E\">#  [ 9 16]]<\/span><\/code><\/pre>\n<p>\u042d\u0442\u043e \u0442\u043e\u0442 \u0441\u0430\u043c\u044b\u0439 \u0440\u0430\u0437\u043d\u043e\u0441 \u0438\u0437 \u043f\u0440\u0435\u0434\u044b\u0434\u0443\u0449\u0435\u0433\u043e \u0440\u0430\u0437\u0434\u0435\u043b\u0430: <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">matrix_power(A, 2)<\/code> &#8212; \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u043c\u0430\u0442\u0440\u0438\u0446\u044b, <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">A ** 2<\/code> &#8212; \u043f\u043e\u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u044b\u0439 \u043a\u0432\u0430\u0434\u0440\u0430\u0442. \u041f\u0443\u0442\u0430\u0442\u044c \u0438\u0445 &#8212; \u0442\u0438\u043f\u0438\u0447\u043d\u0430\u044f \u043e\u0448\u0438\u0431\u043a\u0430 \u043d\u043e\u0432\u0438\u0447\u043a\u0430 \u0432 numpy.<\/p>\n<h2 id=\"s7\">\u041a\u043e\u0440\u043e\u0442\u043a\u043e \u043f\u0440\u043e \u0431\u043e\u043b\u044c\u0448\u0438\u0435 \u0441\u0442\u0435\u043f\u0435\u043d\u0438<\/h2>\n<p>\u0414\u043b\u044f \u0431\u043e\u043b\u044c\u0448\u0438\u0445 n (\u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, A \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 1000) \u0443\u043c\u043d\u043e\u0436\u0430\u0442\u044c \u043c\u0430\u0442\u0440\u0438\u0446\u0443 \u043d\u0430 \u0441\u0435\u0431\u044f \u0442\u044b\u0441\u044f\u0447\u0443 \u0440\u0430\u0437 \u043f\u043e\u0434\u0440\u044f\u0434 \u043d\u0435\u044d\u0444\u0444\u0435\u043a\u0442\u0438\u0432\u043d\u043e. \u0415\u0441\u0442\u044c \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u044f &#8212; <strong>\u0431\u044b\u0441\u0442\u0440\u043e\u0435 \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c<\/strong> (\u0431\u0438\u043d\u0430\u0440\u043d\u043e\u0435): \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u0440\u0430\u0437\u0431\u0438\u0432\u0430\u044e\u0442 \u043d\u0430 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u044b, \u0438 \u0447\u0438\u0441\u043b\u043e \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0439 \u0440\u0430\u0441\u0442\u0435\u0442 \u043a\u0430\u043a \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043e\u0442 n, \u0430 \u043d\u0435 \u043b\u0438\u043d\u0435\u0439\u043d\u043e. \u0423\u0433\u043b\u0443\u0431\u043b\u044f\u0442\u044c\u0441\u044f \u0437\u0434\u0435\u0441\u044c \u043d\u0435 \u0431\u0443\u0434\u0443 &#8212; \u0434\u043b\u044f \u0443\u0447\u0435\u0431\u043d\u044b\u0445 \u043f\u0440\u0438\u043c\u0435\u0440\u043e\u0432 \u0445\u0432\u0430\u0442\u0430\u0435\u0442 \u043f\u043e\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0433\u043e \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u044f, \u0430 \u0432 \u0440\u0435\u0430\u043b\u044c\u043d\u043e\u043c \u043a\u043e\u0434\u0435 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">numpy.linalg.matrix_power<\/code> \u0443\u0436\u0435 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u0442 \u044d\u0444\u0444\u0435\u043a\u0442\u0438\u0432\u043d\u044b\u0439 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u043f\u043e\u0434 \u043a\u0430\u043f\u043e\u0442\u043e\u043c.<\/p>\n<h2 id=\"s8\">\u0412\u044b\u0432\u043e\u0434\u044b<\/h2>\n<ul>\n<li>\u0412\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043c\u0430\u0442\u0440\u0438\u0446\u044b \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c A^n &#8212; \u044d\u0442\u043e n-\u043a\u0440\u0430\u0442\u043d\u043e\u0435 \u043c\u0430\u0442\u0440\u0438\u0447\u043d\u043e\u0435 \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u043c\u0430\u0442\u0440\u0438\u0446\u044b \u043d\u0430 \u0441\u0430\u043c\u0443 \u0441\u0435\u0431\u044f, \u0430 \u043d\u0435 \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043a\u0430\u0436\u0434\u043e\u0433\u043e \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u0430 \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u043e.<\/li>\n<li>\u041e\u043f\u0435\u0440\u0430\u0446\u0438\u044f \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0430 \u0442\u043e\u043b\u044c\u043a\u043e \u0434\u043b\u044f \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043d\u044b\u0445 \u043c\u0430\u0442\u0440\u0438\u0446 \u0438 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044f; \u043f\u043e \u0441\u043e\u0433\u043b\u0430\u0448\u0435\u043d\u0438\u044e A^1 = A, A^0 = E (\u0435\u0434\u0438\u043d\u0438\u0447\u043d\u0430\u044f \u043c\u0430\u0442\u0440\u0438\u0446\u0430).<\/li>\n<li>\u041a\u0430\u0436\u0434\u044b\u0439 \u044d\u043b\u0435\u043c\u0435\u043d\u0442 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0441\u0447\u0438\u0442\u0430\u0435\u0442\u0441\u044f \u043f\u043e \u043f\u0440\u0430\u0432\u0438\u043b\u0443 \u00ab\u0441\u0442\u0440\u043e\u043a\u0430 \u043d\u0430 \u0441\u0442\u043e\u043b\u0431\u0435\u0446\u00bb: \u0441\u0443\u043c\u043c\u0430 \u043f\u043e\u043f\u0430\u0440\u043d\u044b\u0445 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0439 \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043e\u0432 \u0441\u0442\u0440\u043e\u043a\u0438 \u043f\u0435\u0440\u0432\u043e\u0439 \u043c\u0430\u0442\u0440\u0438\u0446\u044b \u0438 \u0441\u0442\u043e\u043b\u0431\u0446\u0430 \u0432\u0442\u043e\u0440\u043e\u0439.<\/li>\n<li>\u0421\u0442\u0430\u0440\u0448\u0438\u0435 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 \u0443\u0434\u043e\u0431\u043d\u043e \u0441\u0442\u0440\u043e\u0438\u0442\u044c \u043f\u043e \u0446\u0435\u043f\u043e\u0447\u043a\u0435: A^3 = A^2 * A, A^4 = A^3 * A, \u043f\u0435\u0440\u0435\u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0443\u0436\u0435 \u043f\u043e\u0441\u0447\u0438\u0442\u0430\u043d\u043d\u044b\u0439 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442.<\/li>\n<li>\u0414\u043b\u044f \u043d\u0430\u0448\u0435\u0439 A = [[1, 2], [3, 4]] \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u0442\u0441\u044f A^2 = [[7, 10], [15, 22]] \u0438 A^3 = [[37, 54], [81, 118]] &#8212; \u0438 \u0440\u0443\u043a\u0430\u043c\u0438, \u0438 \u0447\u0435\u0440\u0435\u0437 numpy.<\/li>\n<li>\u0412 \u043a\u043e\u0434\u0435 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0439\u0442\u0435 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">numpy.linalg.matrix_power<\/code> \u0434\u043b\u044f \u0441\u0442\u0435\u043f\u0435\u043d\u0438 \u043c\u0430\u0442\u0440\u0438\u0446\u044b; \u043e\u043f\u0435\u0440\u0430\u0442\u043e\u0440 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">**<\/code> \u0432\u043e\u0437\u0432\u043e\u0434\u0438\u0442 \u043f\u043e\u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u043e \u0438 \u0434\u0430\u0441\u0442 \u0434\u0440\u0443\u0433\u043e\u0439 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442.<\/li>\n<\/ul>\n<h2 id=\"s9\">\u0413\u0434\u0435 \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f \/ \u0441\u0432\u044f\u0437\u044c \u0441 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u043e\u0439<\/h2>\n<p>\u0421\u0442\u0435\u043f\u0435\u043d\u0438 \u043c\u0430\u0442\u0440\u0438\u0446 &#8212; \u043d\u0435 \u0430\u0431\u0441\u0442\u0440\u0430\u043a\u0442\u043d\u043e\u0435 \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u0435. \u041d\u0430 \u043d\u0438\u0445 \u0434\u0435\u0440\u0436\u0430\u0442\u0441\u044f \u043c\u0430\u0440\u043a\u043e\u0432\u0441\u043a\u0438\u0435 \u0446\u0435\u043f\u0438 (\u043c\u0430\u0442\u0440\u0438\u0446\u0430 \u043f\u0435\u0440\u0435\u0445\u043e\u0434\u043e\u0432 \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 n \u0434\u0430\u0435\u0442 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438 \u0441\u043e\u0441\u0442\u043e\u044f\u043d\u0438\u0439 \u0447\u0435\u0440\u0435\u0437 n \u0448\u0430\u0433\u043e\u0432), \u043f\u043e\u0434\u0441\u0447\u0435\u0442 \u043f\u0443\u0442\u0435\u0439 \u0434\u043b\u0438\u043d\u044b k \u0432 \u0433\u0440\u0430\u0444\u0435 \u0447\u0435\u0440\u0435\u0437 \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u043c\u0430\u0442\u0440\u0438\u0446\u044b \u0441\u043c\u0435\u0436\u043d\u043e\u0441\u0442\u0438, \u0430 \u0442\u0430\u043a\u0436\u0435 \u043c\u043d\u043e\u0433\u0438\u0435 \u0440\u0435\u043a\u0443\u0440\u0440\u0435\u043d\u0442\u043d\u044b\u0435 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0441\u0432\u043e\u0434\u044f\u0442 \u043a \u043c\u0430\u0442\u0440\u0438\u0447\u043d\u043e\u043c\u0443 \u0432\u0438\u0434\u0443 \u0438 \u0443\u0441\u043a\u043e\u0440\u044f\u044e\u0442 \u0431\u044b\u0441\u0442\u0440\u044b\u043c \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435\u043c \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c. \u041f\u0440\u0438\u043a\u043b\u0430\u0434\u043d\u0430\u044f \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u043b\u0438\u043d\u0435\u0439\u043d\u0430\u044f \u0430\u043b\u0433\u0435\u0431\u0440\u0430 &#8212; \u0444\u0443\u043d\u0434\u0430\u043c\u0435\u043d\u0442, \u043d\u0430 \u043a\u043e\u0442\u043e\u0440\u043e\u043c \u0441\u0442\u043e\u044f\u0442 \u043c\u0430\u0448\u0438\u043d\u043d\u043e\u0435 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435, \u043a\u043e\u043c\u043f\u044c\u044e\u0442\u0435\u0440\u043d\u0430\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u0430 \u0438 \u0430\u043d\u0430\u043b\u0438\u0437 \u0434\u0430\u043d\u043d\u044b\u0445.<\/p>\n<div class=\"oj-cta\" style=\"background:#fbf4e4;border-left:4px solid #c9962b;border-radius:8px;padding:16px 20px;margin:24px 0\">\n<p style=\"margin:0 0 8px\"><strong>\u041e\u0441\u0432\u043e\u0439\u0442\u0435 \u0442\u0435\u043c\u0443 \u043d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u0435<\/strong><\/p>\n<p>\u0415\u0441\u043b\u0438 \u0445\u043e\u0447\u0435\u0442\u0441\u044f \u0440\u0430\u0437\u043e\u0431\u0440\u0430\u0442\u044c\u0441\u044f \u0432 \u044d\u0442\u043e\u043c \u0441\u0438\u0441\u0442\u0435\u043c\u043d\u043e, \u0430 \u043d\u0435 \u0441\u043e\u0431\u0438\u0440\u0430\u0442\u044c \u043f\u043e \u043a\u0443\u0441\u043a\u0430\u043c, \u043f\u043e\u0441\u043c\u043e\u0442\u0440\u0438\u0442\u0435 \u043a\u0443\u0440\u0441 <a href=\"https:\/\/otus.ru\/lessons\/prikladnaya-matematika\/?int_article=kak-vozvesti-matricu-v-stepen&amp;int_place=article&amp;int_variant=v1\">\u041f\u0440\u0438\u043a\u043b\u0430\u0434\u043d\u0430\u044f \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0434\u043b\u044f Data Science<\/a> \u0432 Otus &#8212; \u0442\u0430\u043c \u043b\u0438\u043d\u0435\u0439\u043d\u0430\u044f \u0430\u043b\u0433\u0435\u0431\u0440\u0430 \u0434\u0430\u0435\u0442\u0441\u044f \u0441\u0440\u0430\u0437\u0443 \u043f\u043e\u0434 \u0437\u0430\u0434\u0430\u0447\u0438 \u0430\u043d\u0430\u043b\u0438\u0437\u0430 \u0434\u0430\u043d\u043d\u044b\u0445. \u041f\u043e\u043d\u044f\u0442\u044c, \u043f\u043e\u0434\u0445\u043e\u0434\u0438\u0442 \u043b\u0438 \u0444\u043e\u0440\u043c\u0430\u0442, \u043c\u043e\u0436\u043d\u043e \u0431\u0435\u0441\u043f\u043b\u0430\u0442\u043d\u043e: \u0443 Otus \u0440\u0435\u0433\u0443\u043b\u044f\u0440\u043d\u043e \u043f\u0440\u043e\u0445\u043e\u0434\u044f\u0442 <a href=\"https:\/\/otus.ru\/events\/?int_article=kak-vozvesti-matricu-v-stepen&amp;int_place=article&amp;int_variant=v1\">\u043e\u0442\u043a\u0440\u044b\u0442\u044b\u0435 \u0443\u0440\u043e\u043a\u0438 \u0438 \u0432\u0435\u0431\u0438\u043d\u0430\u0440\u044b<\/a>, \u0433\u0434\u0435 \u0442\u0435\u043c\u044b \u0440\u0430\u0437\u0431\u0438\u0440\u0430\u044e\u0442 \u0432\u0436\u0438\u0432\u0443\u044e \u0441 \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u0442\u0435\u043b\u0435\u043c.<\/p>\n<\/div>\n<p>\u0421\u043c\u0435\u0436\u043d\u044b\u0435 \u0442\u0435\u043c\u044b: <a href=\"https:\/\/otus.ru\/journal\/matricy-osnovnye-operacii\/\">\u041c\u0430\u0442\u0440\u0438\u0446\u044b: \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0435 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438<\/a>, <a href=\"https:\/\/otus.ru\/journal\/matricy-i-uravneniya-chto-nuzhno-znat-dlya-ih-resheniya\/\">\u041c\u0430\u0442\u0440\u0438\u0446\u044b \u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f: \u0447\u0442\u043e \u043d\u0443\u0436\u043d\u043e \u0437\u043d\u0430\u0442\u044c \u0434\u043b\u044f \u0438\u0445 \u0440\u0435\u0448\u0435\u043d\u0438\u044f<\/a>.<\/p>\n<h2 id=\"s10\">FAQ<\/h2>\n<p><strong>\u041c\u043e\u0436\u043d\u043e \u043b\u0438 \u0432\u043e\u0437\u0432\u0435\u0441\u0442\u0438 \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u043d\u0435\u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043d\u0443\u044e \u043c\u0430\u0442\u0440\u0438\u0446\u0443 (\u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, 2&#215;3)?<\/strong><br \/>\n\u041d\u0435\u0442. \u0427\u0442\u043e\u0431\u044b \u0443\u043c\u043d\u043e\u0436\u0438\u0442\u044c \u043c\u0430\u0442\u0440\u0438\u0446\u0443 \u043d\u0430 \u0441\u0430\u043c\u0443 \u0441\u0435\u0431\u044f, \u0447\u0438\u0441\u043b\u043e \u0435\u0435 \u0441\u0442\u043e\u043b\u0431\u0446\u043e\u0432 \u0434\u043e\u043b\u0436\u043d\u043e \u0441\u043e\u0432\u043f\u0430\u0434\u0430\u0442\u044c \u0441 \u0447\u0438\u0441\u043b\u043e\u043c \u0441\u0442\u0440\u043e\u043a, \u0430 \u044d\u0442\u043e \u0432\u044b\u043f\u043e\u043b\u043d\u0435\u043d\u043e \u0442\u043e\u043b\u044c\u043a\u043e \u0434\u043b\u044f \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043d\u043e\u0439. \u0414\u043b\u044f \u043c\u0430\u0442\u0440\u0438\u0446\u044b 2&#215;3 \u0437\u0430\u043f\u0438\u0441\u044c A^2 \u043d\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0430.<\/p>\n<p><strong>\u0427\u0442\u043e \u0442\u0430\u043a\u043e\u0435 A \u0432 \u043d\u0443\u043b\u0435\u0432\u043e\u0439 \u0438 \u043f\u0435\u0440\u0432\u043e\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u0438?<\/strong><br \/>\nA^1 &#8212; \u044d\u0442\u043e \u0441\u0430\u043c\u0430 \u043c\u0430\u0442\u0440\u0438\u0446\u0430 A, \u0431\u0435\u0437 \u0438\u0437\u043c\u0435\u043d\u0435\u043d\u0438\u0439. A^0 \u043f\u043e \u0441\u043e\u0433\u043b\u0430\u0448\u0435\u043d\u0438\u044e \u0440\u0430\u0432\u043d\u0430 \u0435\u0434\u0438\u043d\u0438\u0447\u043d\u043e\u0439 \u043c\u0430\u0442\u0440\u0438\u0446\u0435 E \u0442\u043e\u0433\u043e \u0436\u0435 \u0440\u0430\u0437\u043c\u0435\u0440\u0430 (\u0435\u0434\u0438\u043d\u0438\u0446\u044b \u043d\u0430 \u0433\u043b\u0430\u0432\u043d\u043e\u0439 \u0434\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u0438, \u043d\u0443\u043b\u0438 \u0432\u043d\u0435 \u0435\u0435) &#8212; \u043f\u043e \u0430\u043d\u0430\u043b\u043e\u0433\u0438\u0438 \u0441 \u0442\u0435\u043c, \u0447\u0442\u043e \u0447\u0438\u0441\u043b\u043e \u0432 \u043d\u0443\u043b\u0435\u0432\u043e\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 \u0440\u0430\u0432\u043d\u043e 1.<\/p>\n<p><strong>\u041a\u0430\u043a \u0431\u044b\u0442\u044c \u0441 \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u044c\u044e, \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440 A^(-1)?<\/strong><br \/>\n\u041e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u0435\u0442\u0441\u044f \u0447\u0435\u0440\u0435\u0437 \u043e\u0431\u0440\u0430\u0442\u043d\u0443\u044e \u043c\u0430\u0442\u0440\u0438\u0446\u0443: A^(-1) &#8212; \u044d\u0442\u043e \u043e\u0431\u0440\u0430\u0442\u043d\u0430\u044f \u043a A, \u0430 A^(-2) = (A^(-1))^2. \u041d\u043e \u043e\u0431\u0440\u0430\u0442\u043d\u0430\u044f \u043c\u0430\u0442\u0440\u0438\u0446\u0430 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u043d\u0435 \u0434\u043b\u044f \u0432\u0441\u044f\u043a\u043e\u0439 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043d\u043e\u0439 \u043c\u0430\u0442\u0440\u0438\u0446\u044b (\u0442\u043e\u043b\u044c\u043a\u043e \u0435\u0441\u043b\u0438 \u0435\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u0435\u043b\u044c \u043d\u0435 \u0440\u0430\u0432\u0435\u043d \u043d\u0443\u043b\u044e), \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0435 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 &#8212; \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u044b\u0439 \u0441\u043b\u0443\u0447\u0430\u0439 \u0437\u0430 \u0440\u0430\u043c\u043a\u0430\u043c\u0438 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044f.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0412\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043c\u0430\u0442\u0440\u0438\u0446\u044b \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c &#8212; 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