Let P = [ ccc 3 & -1 & -2 2 & 0 & 3 & -5 & 0 ] where R . Suppose … - Vidyadip

Question

Let $P =\left[\begin{array}{ccc}3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0\end{array}\right]$ where $\alpha \in R .$ Suppose $Q =\left[ q _{ ij }\right]$ is a matrix satisfying $PQ = kI _{3}$ for some non-zero $k \in R .$ If $q_{23}=-\frac{k}{8}$ and $|Q|=\frac{k^{2}}{2}$, then $\alpha^{2}+ k ^{2}$ is equal to...........

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Answer

a
$PQ = kI$

$| P | \cdot| Q |= k ^{3}$

$\Rightarrow| P |=2 k \neq 0 \Rightarrow P$ is an invertible matrix

$\because PQ = kI$

$\therefore Q=k P^{-1} I$

$\therefore Q=\frac{\text { adj.P }}{2}$

$\because q _{23}=-\frac{ k }{8}$

$\therefore \frac{-(3 \alpha+4)}{2}=-\frac{ k }{8} \Rightarrow k =4$

$\therefore| P |=2 k \Rightarrow k =10+6 \alpha \ldots( i )$

Put value of $k$ in (i).. we get $\alpha=-1$

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