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 such that $PQ=kI$, where, $k\in R,k\ne 0$ and $I$ is the identity matrix of order $3$. If $q_{23}=-\frac{k}{8}$ and $det\left(Q\right)=\frac{k^2}{2}$, then

• A. $det\left(Q\right(adj\left(P\right)\left)\right)=2^{13}$
• B. $\alpha =0,k=8$
• C. $det\left(P\right(adj\left(Q\right)\left)\right)=2^9$
• D. $4\alpha -k+8=0$

Answer 1

Answer: (C), (D)

$4\alpha -k+8=0$

$PQ=kI$
$\Rightarrow |P||Q|=k^3$
$\Rightarrow |P|\frac{k^2}{2}=k^3$
$\Rightarrow |P|=2k=12\alpha +20$ ... (i)
Also, $Q=k{P}^{-1}$
$=k.\frac{adjP}{|P|}=\frac{adjP}{2}$
So, $q_{23}=-\left(\frac{3\alpha +4}{2}\right)=\frac{-k}{8}$
$\Rightarrow 3\alpha +4=\frac{k}{4}$ ... (ii)
On solving (i) & (ii)
$k=4,\alpha =-1,|P|=8,|Q|=8$
$\Rightarrow 4\alpha -k+8=0$
$|PadjQ|=|P||Q{|}^2=8^3=2^9$
$|QadjP|=|Q||P{|}^2=8^3=2^9$