Problem 4. Let A be any 3 X 3 matrix over F such that A3 = O, the zero matrix, while A2 0. (a) Let 1) E F3 be any nonzero vector such that A21: is nonzero. (Such a vector exists because A2 i 0.) Explain why A21), Av, '0 form a basis for F 3. (b) Let P be the 3 x 3 matrix whose columns are given by A21), Av, 11 in this order. (Note that P is invertible by Part (a).) Compute P'1A2'v and P'lA'v. (c) Compute P '1AP. Conclude that any A with the given hypotheses is always similar to a specic matrix