This is a continuation of Problem 1. Show that if A and B are square matrices such that AB = BA, then
(A + B) 3 = A3 + 3A2B + 3AB2 + B3
and
(A + B) 4 = A4 + 4A3B + 6A2B2 + 4AB3 + B4.
Problem 1
Problem, illustrate ways in which the algebra of matrices is not analogous to the algebra of real numbers.
(a) Suppose that A and B are the matrices of Example 5.
Show that (A + B) 2 ≠ A2 + 2AB + B2.
(b) Suppose that A and B are square matrices such that AB = BA. Show that (A + B) 2 = A2 + 2AB + B2.
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