Question

8. b. -2 3 0 0 0 0 -1 2 0 0-4 0 3 0-2 0 3 0 0 -2 0 3 0 4 o0-1 6 0 0 1 o 2 6 0 0 -1 6 10. For any positive integer k, prove that det(4t) - de(A)*. 11. Prove that if A is invertible, then den(A-1)- I/der(A) - det(4)- 12. We know in general that A-B丰B-A for two n x n matrices. However, prove that: det(A . B)-det(B A). Suppose that X and Y are n x n matrices, where Y is invertible, and we define Z by: Z- Y.X.y-1. Prove that der(X) - det(Z). 13. 14. Suppose that A is a 5 x 5 matrix with det(4)- -60, and Ei through Es are elementary 5 xS matrices obtained from Is in the following ways, respectively: E: exchange rows 2 and 4 of Is E2 : multiply row 5 of, by 7; E: divide row, l of 15 by 3; Ea: add 6 times row 3 of 15 to row 4. Find det(B), where B - E E2.Es Es A 15. Matrices in Block Diagonal Form: Suppose that A1, A2, ..., A are all square matrices, not necessarily of the same size, with k 2. We defined the direct sum of these matrices: A = A1田A2田 田Ak in the Exercises of Section 2.8. Prove that: det(A) - det(A1)det(A)det(A). Hint: Use Induction. Start by showing that if A Ai A2, then det(4) derAi )dert(A).
13 please

Answer 1

Let x and y are n times n matrices and y is invertible implies y^-1 is n times n matrix implies det(y) notequalto 0 Also det(y^-1) = 1/det(y) (1) Now Z = y . x. y^-1 (Given) implies det (z) = det(y .x .y^-1) = det(y). det(x). det(y^-1) = det(y) .det(x).1/det(y) (using (1)) = det(x) Hence det(z) = det(x) Hence proved.