Question

Let $A$ be a non-zero periodic matrix with period $4$ and $A^{12}+B=I,$ where $I$ is identity matrix and $B$ is any square matrix of same order as of $A$ . Matrix product $AB$ is equal to-

• A. $I$
• B. $A$
• C. $A+I$
• D. null matrix

Answer 1

Answer: (D)

$A=A^{4 + 1}\Rightarrow A=A^{5}=A^{9}=A^{13}$
$\therefore A^{12}+B=I$
$A\left(A^{12} + B\right)=A\Rightarrow A^{13}+AB=A\ldots \Rightarrow AB=0$