201 Quiz 5 FR solutions

# Linear Algebra with Applications (3rd Edition)

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Friday (1) Use the row operations 1. (IV - III) 2. (III - II) 3. (II - I) to reduce A to the matrix A 0 = 1 1 1 1 0 3 3 3 0 0 5 5 0 0 0 7 Since adding a row to another row does not change the determinant, det A = det A 0 = 105. (2) (a) False. Here is a counterexample: A = 1 2 4 0 You can check that det B = - 8 = det A . (b) This one is false, but tricky because it is very nearly true. Here is a counterexample: A = 1 0 0 0 0 1 0 1 0 After switching the second and third rows we see that det A = - 1, but the product of the pivots is 1. It is true, however, that the determinant is equal to the product of the pivots up to a sign. (c) False; let B = - A for any n × n invertible matrix A . (d) True. By the product rule for determinants, if the determinant of A or B is zero, so is the determinant of AB . (3) (a) If A is n × n , then det( - A ) = ( - 1) n det A . The matrix for T is - I n . Thus det( T ) = ( - 1) n det I n = ( - 1)

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