Chapter 2.4 Determinants-Applications

If i j form a matrix a by replacing row j of a with

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Unformatted text preview: i = j , form a matrix A by replacing row j of A with another copy of row i . Since A has proportional rows, n 0 = det A = n aik Cik = k =1 ajk Cik . k =1 Outline The Classical Adjoint A Formula for the Inverse of a Matrix A Formula for the Inverse of a Matrix Theorem If A is invertible, then A−1 = 1 adj(A). det A Cramer’s Rule Eigenvalues Outline The Classical Adjoint A Formula for the Inverse of a Matrix Cramer’s Rule The Proof Proof. Set B = adj(A) = [Cij ]T and AB = [cij ]. For each 1 ≤ i , j ≤ n, n cij = aik Cjk = k =1 det A if i = j 0 if i = j Hence, 1 cij = det A so 1 det A AB 1 if i = j 0 if i = j = In . We conclude that A−1 = 1 det A adj(A). Eigenvalues...
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This note was uploaded on 02/10/2014 for the course MATH 2270 taught by Professor Kenyonj.platt during the Spring '14 term at Snow College.

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