\n
If for a given square matrix $A$ there exists a matrix $B$ such that $AB = I$ , then $B$ is called an inverse<\/em> of $A$ and is denoted by $A^{-1}$ . The following theorem is fundamental. <\/p>\n
11. If $A$ is a non-singular square matrix of order n<\/em> [i.e. $\\det(A) \\ne 0$ ], then there exists a unique inverse $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$ and we can express $A^{-1}$ in the following form <\/p>\n
$\\displaystyle A^{-1} = \\frac{(A_{jk})^T}{\\det(A)} \\cdots(14)$ <\/p>\n
where $(A_{jk})$ is the matrix of cofactors $A_{jk}$ and $(A_{jk})^T = (A_{kj})$ is its transpose. <\/p>\n
The following express some properties of the inverse: <\/p>\n
$(AB)^{-1} = B^{-1}A^{-1} ,\\ (A^{-1})^{-1} = A \\cdots(15)$ <\/p>\n<\/blockquote>\n
<\/iframe><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>If for a given square matrix there exists a matrix such that , then is called an inverse of and is denoted by … <a href=\"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5236\" class=\"more-link\"><span class=\"screen-reader-text\">“Inverse of a matrix” \u306e<\/span>\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":1,"featured_media":6040,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_crdt_document":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[9],"tags":[],"class_list":["post-5236","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/5236","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5236"}],"version-history":[{"count":5,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/5236\/revisions"}],"predecessor-version":[{"id":5424,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/5236\/revisions\/5424"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/media\/6040"}],"wp:attachment":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5236"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5236"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5236"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}