In $P$ and $Q$ are square matrices such that $P^{2006}=0$ and $P Q=P+Q$, then $\operatorname{det}(Q)$ will be

Question

In $P$ and $Q$ are square matrices such that $P^{2006}=0$ and $P Q=P+Q$, then $\operatorname{det}(Q)$ will be, 0, 1 only, -1 only, \pm 1

MARKS App · Accepted Answer

$\begin{array}{llll} & P^{2006} =O \text { and } P Q=P+Q \ \Rightarrow & P^{2006} Q =P^{2006}+Q \cdot P^{2005} \ \Rightarrow & O =O+Q \cdot P^{2005} \ \Rightarrow & P^{2005} \cdot Q =O \ \Rightarrow & \operatorname{det} \cdot\left(P^{2005} \cdot Q\right) =O \ \Rightarrow & \operatorname{det} P^{2005}(\operatorname{det} Q) =O \Rightarrow \operatorname{det} Q=O \end{array}$

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