If $A$ and $B$ are $n \times n$ square matrices such that $(2 A+B)^2+(A-3 B)^2=5 A^2-2 A B+10 B^2$, then $A…
If $A$ and $B$ are $n \times n$ square matrices such that $(2 A+B)^2+(A-3 B)^2=5 A^2-2 A B+10 B^2$, then $A B A B=$
$\frac{1}{2}\left[(A-B)^2+(A+B)^2\right]$
$4 A B$
$\frac{1}{2}\left[(A+B)^2-(A-B)^2\right]$
$A^2 B^2$
Solution
Given, $(2 A+B)^2+(A-3 B)^2=5 A^2-2 A B+10 B^2$
$\Rightarrow 4 A^2+B^2+2 A B+2 B A+A^2+9 B^2$
$-3 A B-3 B A-5 A^2+2 A B-10 B^2=0$
$\Rightarrow \quad A B-B A=0 \Rightarrow A B=B A$
Then, $A B A B=(A B)(A B)=(A A)(B B)=A^2 B^2$