{"id":5236,"date":"2014-04-21T06:05:54","date_gmt":"2014-04-20T21:05:54","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5236"},"modified":"2014-08-01T18:33:30","modified_gmt":"2014-08-01T09:33:30","slug":"inverse-of-a-matrix","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5236","title":{"rendered":"Inverse of a matrix"},"content":{"rendered":"
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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