Lecture18 - University of Toronto at Scarborough Department of Computer& Mathematical Sciences MAT A23 Winter 2008 Lecture 18 Cramer’s Rule

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Unformatted text preview: University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT A23 Winter 2008 Lecture 18 Cramer’s Rule Sophie Chrysostomou Cramer’s Rule theorem 0.1 If Ax = b is a system of n linear equations in n unknowns and det(A) = 0, then the unique solution x = [x1 , x2 , · · · , xn ] is of the form xk = det(Bk ) det(A) for k = 1, · · · n where Bk is the matrix A with the k th column replaced by the column vector b. example 0.2 Use Cramer’s Rule to solve the system: x1 + x2 + x3 = 0 2x1 − x2 = 11 x2 + 4x3 = 3 1 definition 0.3 Let A be an n × n matrix. 1. Let Aij denote the (n − 1) × (n − 1) matrix obtained by removing the ith row and j th column of A. 2. The cofactor of aij of A is aij = (−1)i+j det(Aij ) = (−1)i+j |Aij | 3. Let A = [ aij ] be the matrix with ij th entry the ij th cofactor of A. Then the adjoint of A denoted by adj (A) is the n × n matrix adj (A) = (A )T . theorem 0.4 If A is an n × n matrix then (adj (A)) A = A (adj (A)) = det(A) 2 corollary 0.5 If A is an n × n matrix and det(A) = 0, then A− 1 = 1 adj (A). det(A) 3 ...
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This note was uploaded on 05/22/2010 for the course MATH MATA23 taught by Professor Sophie during the Spring '10 term at University of Toronto- Toronto.

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