Stitz-Zeager_College_Algebra_e-book

Using test points 0 0 and 0 3 we nd that our solution

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Unformatted text preview: we’re trying to say easier to follow. 516 Systems of Equations and Matrices 9 C11 C21 C31 −9 −2 7 13 1 1 C12 C22 C32 = − 10 8 −15 = − 10 = 13 det(A) 13 2 C13 C23 C33 2 −1 −3 − 13 A−1 2 13 8 − 13 1 13 7 − 13 15 13 3 13 To generalize this to invertible n × n matrices, we need another definition and a theorem. Our definition gives a special name to the cofactor matrix, and the theorem tells us how to use it along with det(A) to find the inverse of a matrix. Definition 8.15. Let A be an n × n matrix, and Cij denote the ij cofactor of A. The adjoint of A, denoted adj(A) is the matrix whose ij -entry is the ji cofactor of A, Cji . That is C11 C21 . . . Cn1 C12 C22 . . . Cn2 adj(A) = . . . . . . . . . C1n C2n . . . Cnn This new notation greatly shortens the statement of the formula for the inverse of a matrix. Theorem 8.9. Let A be an invertible n × n matrix. Then A−1 = 1 adj(A) det(A) For 2 × 2 matrices, Theorem 8.9 reduces to a fairly simple formula. Equation 8.2. For an invertible 2 × 2 matrix, ab cd −1 = 1 ad − bc d −b −c a The p...
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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