QJEEM23Maf853: JEE Main 2023 Mathematics - Class 12 JEEM23Maf853 Solution: If A and B are two non-zero n n matrices such that , then :

Question

Verified Solution • 2025–26 Session moderate

If A and B are two non-zero n $\times$ n matrices such that $\mathrm{A^2+B=A^2B}$ , then :

Accepted Answer

Given :

A^{2}+B=A^{2} B\quad...(i)

\Rightarrow A^{2}+B-I=A^{2} B-I

\Rightarrow A^{2} B-A^{2}-B+I=I

\Rightarrow A^{2}(B-I)-I(B-I)=I

\Rightarrow\left(A^{2}-I\right)(B-I)=I

\therefore A^{2}-I

is the inverse matrix of

B-I

and vice versa.

So,

(B-I)\left(A^{2}-I\right)=I

\Rightarrow B A^{2}-B-A^{2}+I=I

\therefore A^{2}+B=B A^{2} \quad...(ii)

So, by (i) and (ii)

A^{2} B=B A^{2}

Step-by-Step Verified Solution