Question

If $A$ and fl are square matrices of size $n \times n$ such that $A^{2}-B^{2}=\left(A-B\right)\left(A+B\right)$, then which of the following will be always true ?

• A. $AB = BA$
• B. either of $A$ or $B$ is a zero matrix
• C. either of $A$ or $B$ is an identity matrix
• D. $A = B$

Answer 1

Answer: (A)

$AB = BA$

Answer 2

$\because A^{2}-B^{2}=\left(A-B\right)\left(A+B\right)$ $\Rightarrow A^{2}-B^{2}=A^{2}-B^{2}+AB-BA$ $\Rightarrow AB=BA$