Search any question & find its solution
Question:
Answered & Verified by Expert
$A$ and $B$ are two matrices such that $A B=B$, $B A=A$, then $A^{2}+B^{2}=$
Options:
Solution:
1765 Upvotes
Verified Answer
The correct answer is:
$A+B$
Given, $A B=B$
Multiply by $A$ on both sides
$$
A B A=B A \quad \text{...(i)}
$$
Also, $B A=A$
Multiply by $B$ on both sides
$$
B A B=A B \quad \text{...(ii)}
$$
Adding Eqs. (i) and (ii), we get
$A B A+B A B=B A+A B$
$\Rightarrow \quad A(B A)+B(A B)=B A+A B$
$\Rightarrow \quad A A+B B=A+B$
$\Rightarrow \quad A^{2}+B^{2}=A+B$
Multiply by $A$ on both sides
$$
A B A=B A \quad \text{...(i)}
$$
Also, $B A=A$
Multiply by $B$ on both sides
$$
B A B=A B \quad \text{...(ii)}
$$
Adding Eqs. (i) and (ii), we get
$A B A+B A B=B A+A B$
$\Rightarrow \quad A(B A)+B(A B)=B A+A B$
$\Rightarrow \quad A A+B B=A+B$
$\Rightarrow \quad A^{2}+B^{2}=A+B$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.