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3 and 4 .Determinants and Matrices
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Let $P=\left[\begin{array}{ccc}3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0\end{array}\right]$, where $\alpha \in \mathbb{R}$. Suppose $Q=\left[q_{i j}\right]$ is a matrix such that $P Q=k I$, where $k \in \mathbb{R}, k \neq 0$ and $I$ is the identity matrix of order $3$ . If $q_{23}=-\frac{k}{8}$ and $\operatorname{det}(Q)=\frac{k^2}{2}$, then
($A$) $\quad \alpha=0, k=8$
($B$) $4 \alpha-k+8=0$
($C$) $\operatorname{det}(P \operatorname{adj}(Q))=2^9$
($D$) $\operatorname{det}(Q \operatorname{adj}(P))=2^{13}$
A
$B,C$
B
$B,D$
C
$B,A$
D
$B,C,A$
(IIT-2016)
Solution
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Standard 12
Mathematics
