Let A be a non-zero periodic matrix with period 4 and A12+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-

Question

Let A be a non-zero periodic matrix with period 4 and A12+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

Tardigrade Admin · Accepted Answer

A=A4 + 1⇒ A=A5=A9=A13 ∴ A12+B=I A(A12 + B)=A⇒ A13+AB=A ldots ⇒ AB=0

Q. 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 $A B$ is equal to-

$I$

$A$

$A + I$

null matrix

$A = A^{4 + 1} \Rightarrow A = A^{5} = A^{9} = A^{13}$
$∴ A^{12} + B = I$
$A (A^{12} + B) = A \Rightarrow A^{13} + A B = A \dots \Rightarrow A B = 0$

Q. Let A be a non-zero periodic matrix with period 4 and A12+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-

I

A

A+I

null matrix

A=A4+1⇒A=A5=A9=A13 ∴A12+B=I A(A12+B)=A⇒A13+AB=A…⇒AB=0

Q. 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 $A B$ is equal to-

$I$

$A$

$A + I$

$A = A^{4 + 1} \Rightarrow A = A^{5} = A^{9} = A^{13}$
$∴ A^{12} + B = I$
$A (A^{12} + B) = A \Rightarrow A^{13} + A B = A \dots \Rightarrow A B = 0$