College Algebra Exam Review 354

College Algebra - n matrix over R Let A i;j be the.n ± 1 –by–.n ± 1 matrix obtained by deleting the i –th row and the j –column of A The

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364 8. MODULES Lemma 8.3.12. If A is a k –by– k matrix, and E ` is the ` –by– ` identity matrix, then det ± A 0 0 E ` ² D det ± E ` 0 0 A ² D det .A/: Proof. The function ±.A/ D det ± A 0 0 E ` ² is alternating and multilinear on the columns of A , and therefore by Corollary 8.3.8 , ±.A/ D det .A/±.E k / . But ±.E k / D det .E k C ` / D 1: This shows that det ± A 0 0 E ` ² D det .A/ . The proof of the other equality is the same. n Lemma 8.3.13. If A and B are square matrices, then det ± A 0 C B ² D det .A/ det .B/: Proof. We have ± A 0 C B ² D ± A 0 0 E ²± E 0 C E ²± E 0 0 B ² : Therefore, det ± A 0 C B ² is the product of the determinants of the three ma- trices on th right side of the equation, by Corollary 8.3.9 (d). According to the previous lemma det ± A 0 0 E ² D det .A/ and det ± E 0 0 B ² D det .B/ . Finally ± E 0 C E ² is triangular with 1 ’s on the diagonal, so its determinant is equal to 1 , by Corollary 8.3.9 (c). n Let A be an n –by–
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Unformatted text preview: n matrix over R . Let A i;j be the .n ± 1/ –by– .n ± 1/ matrix obtained by deleting the i –th row and the j –column of A . The de-terminant det .A i;j / is called the .i;j/ minor of A , and . ± 1/ i C j det .A i;j / is called the .i;j/ cofactor of A . The matrix whose .i;j/ entry is . ± 1/ i C j det .A i;j / is called the cofactor matrix of A . The transpose of the cofactor matrix is sometimes called the adjoint matrix of A , but this terminology should be avoided as the word adjoint has other incompatible meanings in linear algebra. The following is called the cofactor expansion of the determinant....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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