Question

If A and B are both n × n matrices of real numbers, determine if each of the following is necessarily true. That is, will the relationships ALWAYS be true regardless of the specific values in A and B. Prove this using the rules of matrix arithmetic as applied to A and B. Do not use counter examples or specific matrices. Show enough steps that it is clear why you have reached your conclusion.

(a) (A + B)^2 ____ A^2 + 2AB + B^2
(b) ABA ____ A^2 B
(c) ABAB ___ (AB)2

Answer 1

(a) False

(b) Fase

(c) True

Explanation:

(a)
`(A + B) ^ 2 = (A + B) (A + B) = AA + AB + BA + BB = A ^ 2 + AB + BA + B ^ 2`. Since AB does not necessarily equal BA, we cannot say that
`AB + BA = 2AB` or
`AB + BA = 2BA.`

(b) Since AB does not necessarily equal BA, we cannot say that
`ABA = AAB = A ^ 2B` or
`ABA = BAA = BA ^ 2 = A ^ 2B.`.

(c) Since
`ABAB = (AB) (AB) = (AB) ^ 2`, then we can state that
`ABAB = (AB) ^ 2`.