7. (15 pts) Let A and B ben x n square matrices such that AB = BA. (a) Suppose A is an eigenvalue of A such that dim EA

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7 15 Pts Let A And B Ben X N Square Matrices Such That Ab Ba A Suppose A Is An Eigenvalue Of A Such That Dim Ea 1 (32.59 KiB) Viewed 19 times

7. (15 pts) Let A and B ben x n square matrices such that AB = BA. (a) Suppose A is an eigenvalue of A such that dim EA (A) = 1. Let v be an eigenvector of A with eigenvalue A. Show that v is also an eigenvector of B. (b) Suppose A1, A2,..., An are n distinct eigenvalues of A and dim EA (₁) = 1 for i = 1,..., n. Show that B is diagonalizable. (c) Let n = 3. Find two 3 x 3 square matrices A, B, satisfying all the following conditions i. AB = BA. ii. A is not a diagonal matrix. iii. A is diagonalizable iv. B is not diagonalizable. Justify your answer. Warning: No point for just writing down A, B without explanation.