A square matrix is an upper triangular matrix if all elements below the principal diagonal are zero. So a 2 × 2 upper triangular matrix has the form

where a, b and d are real numbers. Discuss the validity of each of the following statements. If the statement is always true, explain why. If not, give examples.
(A) If A and B are 2 × 2 upper triangular matrix, then A + B is a 2 × 2 upper triangular matrix.
(B) If A and B are 2×2 upper triangular matrix, then A + B = B + A.
(C) If A and B are 2 × 2 upper triangular matrix, then AB is a 2 × 2 upper triangular matrix.
(D) If A and B are 2×2 upper triangular matrix, then AB = BA.

0 d