Question

If $A$ and $B$ are square matrices of same order satisfying $A+B=AB,$ then

• A. $AB=BA$
• B. $AB=-BA$
• C. $A+B=-BA$
• D. $A-B=BA$

Answer 1

The correct option is A $AB=BA$

Given, $A+B=AB\cdots \left(1\right)$
$\Rightarrow AB-A-B+I=I$
$\Rightarrow (I-A)(I-B)=I$
$\Rightarrow I-A$ and $I-B$ are inverse of each other.
So, $(I-B)(I-A)=I$
$\Rightarrow I-A-B+BA=I$
$\Rightarrow A+B=BA\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right),$
$AB=BA$