Q43 / 160 · AP EAMCET 2022 (08 Jul Shift 2) · Mathematics

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 $A B A B=$

  • A
    $\frac{1}{2}\left[(A-B)^2+(A+B)^2\right]$
  • B
    $4 A B$
  • C
    $\frac{1}{2}\left[(A+B)^2-(A-B)^2\right]$
  • Correct answer
    D
    $A^2 B^2$

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, $A B A B=(A B)(A B)=(A A)(B B)=A^2 B^2$

Hence, the correct answer is option: 'D'.

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