Question

Suppose A and B are square matrices of the same size. Which of the following are necessarily true?

a. (A-B)^2=A^2-2AB + B^2
b. (AB)^2=A^2B^2
c. (A + B)^2-A^2 + 2AB + B^2

Answer 1

Answer:

c. (A + B)^2-A^2 + 2AB + B^2

Step-by-step explanation:

Given that:

a.

(A-B)^2 = A^2 - 2AB + B^2

If and only if AB = BA

Then;

(A-B)^2 = (A -B ) (A - B)

(A-B)^2 = A^2 - AB-BA + B^2    (FALSE)

b.

(AB)^2=A^2B^2

on true if any only if AB =BA

(AB)^2= (AB) (AB)

c.

(A+ B)^2 =  A^2 + 2AB + B^2

(A+ B)^2 = (A + B) (A+B)

(A+ B)² = A × A + A × B + B × A + B × B

(A+ B)^2 = A^2 + A*B + B*A + B^2

This is true