Question

If $A,B,C$ are square matrices of same order such that $AB=O$ and $BC=I$ then ${(A+B)}^2{(A+C)}^2=$

• A. $O$
• B. $2I$
• C. $I$
• D. $3I$

Answer 1

The correct option is C $I$

Given :

$AB=0$ (Zero Matrix)

$\Rightarrow A=0$ or $B=0⟶\left(1\right)$

Also $BC=I$

$\Rightarrow B=C^{-1}$ or $C=B^{-1}⟶\left(2\right)$

$\Rightarrow B\ne 0⟶\left(3\right)$

Hence from $\left(1\right)$ & $\left(3\right)$ we have $A=0$

$\therefore$ ${(A+B)}^2{(A+C)}^2={(O+B)}^2{(O+C)}^2=B^2{C}^2={\left(BC\right)}^2=I^2=II$

$\therefore$ Correct answer : Option C.