FinalTestReviewSpring2011

pts solution deta i 0 0 0 we have two eigenvalues

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Unformatted text preview: g exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. (a) State the Cofactor Formula for finding the inverse A−1 = [xij ] of an n × n matrix A = [aij ]. C ji (?? pts) Solution: xij = det(A) , where Cji = (−1)j +i det(Mji ) and Mji is obtained from A by removing its j th row and ith column. 1200 3 4 0 0 (b) Use the Cofactor Formula you citated in (a) to find x21 , if A = 0 0 5 0 . 0007 ￿ ￿ ￿ ￿ ￿1 2 0 0￿ ￿1 2 0 0￿ ￿ ￿ ￿ ￿ ￿ 3 4 0 0 ￿ −3￿1 ￿ 0 −2 0 0 ￿ r ￿ ￿ ￿ ￿ = −2 · 35. (?? pts) Solution: det(A) = ￿ =￿ 0 0 5 0￿ 0 0 5 0￿ ￿ ￿ ￿ ￿ ￿0 0 0 7￿ ￿0 0 0 7￿ ￿ ￿ ￿3 0 0￿ ￿ ￿ C12 C12 = (−1)1+2 ￿ 0 5 0 ￿ = −3 · 35. So, x21 = det(A) = −3··35 = 1.5 −2 35 ￿ ￿ ￿0 0 7￿ 1 134 Ex. 2. Find all eigenvalues and associated eigenvectors of the matrix A = 0 0 1 . 000 Is matrix A diagonalizable? ￿ ￿ ￿ 1−λ 3 4￿ ￿ ￿ −λ 1 ￿ = (1 −...
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