Linear algebra questions
Math 220 Summer Written Homework 6 1−h 6 1. Let A = . 5 2−h (a) Find the determinant of A. (b) Find all values of h for which A is invertible. 2. In each of the following, the matrix is an elementary matrix. State which row operation the elementary matrix corresponds to, and then state the determinant of the elementary matrix. 1 0 0 0 1 0 −5 0 0 (a) E = 0 1 0 (b) E = 1 0 0 (c) E = 0 1 0. 0 −5 1 0 0 1 0 0 1 a b c 3. Suppose B = d e f and that det(B) = 5. Compute the following determinants: g h i a b c a b c a b c d e f (a) (b) 2d 2e 2f (c) 4g 4h 4i g + 2a h + 2b i + 2c 4g 4h 4i 2d 2e 2f 4. Find the determinant using row reduction to echelon form. −2 −8 7 −1 −4 5 1 5 −4 2 4 0 −2 3 1 2 4 (a) (b) 5 10 3 −6 2 4 0 −3 −2 3 2 5. Use a determinant to show that the set of vectors 3 , 1 , 2 is 1 3 −1 linearly dependent. Your justification must include at least one complete sentence explaining why the set is linearly dependent in addition to your determinant calculation. 6. Let A, B be a 3 × 3 matrices with det(A) = 7 and det(B) = 3. Find (a) det(2AT ) (c) det(A−1 ) (e) det((2A)−1 ) (d) det(2A−1 ) (f) det((AB)−1 ) 2 0 0 7. Suppose B = 0 0 1 . In each part determine the corresponding 3×3 elemen0 1 1 tary matrix E. Then compute EB and verify that EB is the matrix that results in performing the corresponding row operation to B. (b) det (2AB) 1 (a) R1 ↔ R2 (b) 12 R1 (c) R3 → R3 − R1 2 0 0 8. Suppose B = 0 0 1 . 0 1 1 (a) Perform row operations to reduce B to In , keep track of the corresponding elementary matrices, E1 , E2 , …. (b) Write out appropriate product of the Ei′ s to determine B −1 . you can leave your answer as the product and not actually compute it. 2
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