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Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants1 Mark
MCQ
Let $\text{P}$ and $\text{Q}$ be $3\times3$ matrices with $\text{P}\neq\text{Q}.$ If $\text{P}^3=\text{Q}^3$ and $\text{P}^2\text{Q}=\text{Q}^2\text{P}$ then determinant of $(\text{P}^2+\text{Q}^2)$ is equal to:
  • A
    $-2$
  • B
    $1$
  • ✓
    $0$
  • D
    $-1$
✓

Answer

Correct option: C.
$0$
$\text{P}^3=\text{Q}^3$
$\Rightarrow\text{P}^3- \text{P}^2\text{Q}=\text{Q}^3- \text{Q}^2\text{P}$
$\Rightarrow\text{P}^2(\text{P- Q})=\text{Q}^2(\text{Q- P})$
$\Rightarrow \text{P}^2(\text{P- Q})-\text{Q}^2(\text{Q- P})=0$
$\Rightarrow (\text{P}^2+\text{Q}^2)(\text{P}-\text{Q})=0$
$\Rightarrow|\text{P}^2+\text{Q}^2|=0$

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