If A and B are square matrices of size n × n such that A2-B2=(A-B)(A+B), Then which of the following will be always true?

Question

If A and B are square matrices of size n × n such that A2-B2=(A-B)(A+B), Then which of the following will be always true? (A) either of A or B is a zero matrix (B) either of A or B is an identity matrix (C) A=B (D) A B=B A

Tardigrade Admin · Accepted Answer

A2-B2=(A-B)(A +B) ⇒ A2-B2=A2+A B-B A-B2 ⇒ A B-B A=O ⇒ A B=B A

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

either of A or B is a zero matrix

either of A or B is an identity matrix

A=B

A B=B A

$A^{2} - B^{2} = (A - B) (A + B)$
$\Rightarrow A^{2} - B^{2} = A^{2} + A B - B A - B^{2}$
$\Rightarrow A B - B A = O$
$\Rightarrow A B = B A$

Q. If A and B are square matrices of size n×n such that A2−B2=(A−B)(A+B), Then which of the following will be always true?

either of A or B is a zero matrix

either of A or B is an identity matrix

A=B

A B=B A

A2−B2=(A−B)(A+B) ⇒A2−B2=A2+AB−BA−B2 ⇒AB−BA=O ⇒AB=BA

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

$A^{2} - B^{2} = (A - B) (A + B)$
$\Rightarrow A^{2} - B^{2} = A^{2} + A B - B A - B^{2}$
$\Rightarrow A B - B A = O$
$\Rightarrow A B = B A$