M 340L HW 8 Solutions

M 340L HW 8 Solutions - cruz(fmc326 HW08 gilbert(56540 This...

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cruz (fmc326) – HW08 – gilbert – (56540) 1 This print-out should have 10 questions. Multiple-choice questions may continue on the next column or page – Fnd all choices before answering. 001 10.0 points Evaluate the expression E = 2 v v v v 4 1 3 3 v v v v 3 v v v v 4 3 1 3 v v v v . 1. E = 72 2. E = 75 correct 3. E = 73 4. E = 74 5. E = 71 Explanation: ±or a 2 × 2 determinant, v v v v a b c d v v v v = ad bc. Thus E = 2 v v v v 4 1 3 3 v v v v 3 v v v v 4 3 1 3 v v v v = 2( 12 3) 3(12 + 3) . Consequently, E = 75 . keywords: matrix, determinant 002 10.0 points The determinant of an n × n matrix A can be deFned recursively in terms of the deter- minants of ( n 1) × ( n 1) submatrices of A . True or ±alse? 1. ±ALSE 2. TRUE correct Explanation: The Minor M jk of an n × n matrix A is the determinant M jk = det[ A jk ] of the ( n 1) × ( n 1) submatrix A jk obtained by deleting the j th -row and k th -column from A , while the Co-factor C jk is deFned by C jk = ( 1) j + k det[ A jk ] = ( 1) j + k M jk . The determinant of A can then be deFned in terms of Co-factors by expanding either along a row as in det[ A ] = a j 1 C j 1 + a j 2 C j 2 + ... + a jn C jn , or down a column as in det[ A ] = a 1 k C 1 k + a 2 k C 2 k + + a nk C nk . Consequently, the statement is TRUE . 003 10.0 points The determinant of a triangular matrix is always the sum of the entries on the main diagonal. True or ±alse?
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