Show that if A is a 2 × 2 matrix such that AB = BA whenever B is a 2 × 2 matrix, then A = cI. where c is a real number and I is the 2 × 2 identity matrix.

SOLUTIONStep 1In this problem, we are asked to show that the matrix whenever AB=BA.Step 2We know that if AB=BA then A and B are simultaneously diagonalizable.This implies that there exist a nonsingular matrix P such that are diagonal matrices.Therefore is a diagonal matrix.Step 3Now we have to show that in all the diagonal elements are same.To show this let us use matrix S[i,j] such that and and all other .Therefore...