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

Answer:

(a) False

(b) Fase

(c) True

Step-by-step 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 $ABAB = (AB) ^ 2$.