College Algebra Exam Review 356

College Algebra Exam Review 356 - (a An element of Mat n.R...

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366 8. MODULES n Corollary 8.3.15. Let A be an n –by– n matrix over R and let C denote the cofactor matrix of A . Then AC t D C t A D det .A/E; where E denotes the identity matrix. Proof. The sum n X j D 1 . ± 1/ i C j a k;j det .A i;j /: is the .k;i/ entry of AC t . Proposition 8.3.14 says that this entry is equal to 0 if k ¤ i and equal to det .A/ if k D i , so AC t D det .A/E . The other equality C t A D det .A/ follows from some gymnastics with transposes: We have .A t / i;j D .A j;i / t . Therefore, . ± 1/ i C j det ..A t / i;j / D . ± 1/ i C j det ..A j;i / t / D . ± 1/ i C j det .A j;i /: This says that the cofactor matrix of A t is C t . Applying the equality al- ready obtained to A t gives A t C D det .A t /E D det .A/E; and taking transposes gives C t A D det .A/E: n Corollary 8.3.16.
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Unformatted text preview: (a) An element of Mat n .R/ is invertible if, and only if, its determi-nant is a unit in R . (b) If an element of Mat n .R/ has a left inverse or a right inverse, then it is invertible Proof. We have already seen that the determinant of an invertible matrix is a unit (Corollary 8.3.9 (e)). On the other hand, if det .A/ is a unit in R , then det .A/ ± 1 C t is the inverse of A . If A has a left inverse, then its determinant is a unit in R , so A is invertible by part (a). n...
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