step by step explanations for each operation please

Example 3.3.11 If 23 =52 for every eigenvalue of the diagonalizable matrix A, show that A3 = 5A. Solution. Let P-AP = D = diag (21, ..., An). Because 23 = 52; for each i, we obtain D3 = diag (23, ..., a,) = diag (521, ..., 52n) = 5D Hence A3 = (PDP-1)3 = PD3p-1 = P(5D)p-1 = 5(PDP-1) = 5A using Theorem 3.3.1. This is what we wanted