Question

Let $P= \begin{bmatrix}3&-1&-2\\ 2&0&\alpha\\ 3&-5&0\end{bmatrix}$, where $\alpha \in R$. Suppose $Q = [q_{ij}]$ 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 ________.

Answer 1

$PQ = kI$
$| P | \cdot| Q |= k ^{3}$
$\Rightarrow \mid P| =2 k \neq 0$
$ \Rightarrow P$ is an invertible matrix
$\because PQ = kI$
$\therefore Q = kP ^{-1} I$
$\therefore Q =\frac{\text { adj.P }}{2}$
$\because q _{23}=-\frac{ k }{8}$
$\frac{-(3 \alpha+4)}{2}=-\frac{ k }{8}$
$ \Rightarrow k =4$
$\therefore | P |=2 k$
$ \Rightarrow k =10+6 \alpha \ldots( i )$
Put value of $k$ in (i).. we get $\alpha=-1$