Question

If $A$ and $B$ are $2\times 2$ matrices, then which of the following is true?

• A. $(A + B)^{2} = A^{2} + B^{2} + 2AB$
• B. $ (A - B)^{2} = A^{2} + B^{2} - 2AB $
• C. $(A - B)(A + B) = A^{2} + AB-BA-B^{2}$
• D. $(A + B)(A - B) = A^{2} - B^{2}$

Answer 1

Answer: (C)

Given that, $A$ and $B$ are $2\times2$ matrices. $\therefore \left(A-B\right)\times\left(A+B\right) = A\times A +A\times B -B\times A - B\times B$
$ = A^{2} - B^{2} + AB -AB$