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The points on the line y x 2 do not satisfy the

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Unformatted text preview: st row.) Cramer’s Rule tells us x11 = det (A1 ) C11 det (A2 ) C12 det (A3 ) C13 = , x21 = = , x31 = = det(A) det(A) det(A) det(A) det(A) det(A) So the first column of the inverse matrix X is: C11 det(A) C C 1 11 12 = C12 = det(A) det(A) C13 C13 det(A) x11 x21 x31 Notice the reversal of the subscripts going from the unknown to the corresponding cofactor of A. This trend continues and we get C x13 C x12 1 31 1 21 x23 = x22 = C22 C32 det(A) det(A) C23 x33 C33 x32 Putting all of these together, we have obtained a new C 1 11 C12 A−1 = det(A) C13 and surprising formula for A−1 , namely C21 C31 C22 C32 C23 C33 To see that this does indeed yield A−1 , we find all of the cofactors of A C11 = −9, C21 = −2, C31 = 7 C12 = 10, C22 = 8, C32 = −15 C13 = 2, C23 = −1, C33 = −3 And, as promised, 7 In a solid Linear Algebra course you will learn that the properties in Theorem 8.7 hold equally well if the word ‘row’ is replaced by the word ‘column’. We’re not going to get into column operations in this text, but they do make some of what...
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