Consider the matrix 1 A = -1 0 Use induction on n to show that for all integers n >1 we have An = I2, where Iz is the 2 x 2 identity matrix.

[ 2n + 1 7][ 6 4) Determine the following matrix product, En +1 33n – 1 n + 1] 2 in terms of n. 5) Let n be a positive integer such that 7| (82-1). Prove that 7| (8-1 - 1). (Hint: Rewrite 8n*1 -1, where a portion of the expression is 8 times 8" - 1.)

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|a Prove that the matrix A = is diagonalizable if –4bc (a – d)². What happens when –4bc = (a –d)² ? b) Give an example of a 3 × 3 matrix that you know is not diagonalizable and explain why this is so.