Let $P=\left[\begin{array}{rrr}3& -1& -2\\ 2& 0& \alpha \\ 3& -5& 0\end{array}\right],$ where $\alpha \in R.$ Suppose $Q=\left[q_{ij}\right]$ is a matrix satisfying $PQ=k{I}_3$ for
some non-zero $k\in R.$ If $q_{23}=-\frac{k}{8}$ and $\left|Q\right|=\frac{k^2}{2}$, then ${\alpha }^2+k^2$ is equal to_________.

$kx+y+2z=1$

$3x-y-2z=2$

$-2x-2y-4z=3$

has infinitely many solutions, then $k$ is equal to ______ .

Let $A$ be a $2\times 2$ real matrix with entries from $\left\{0, 1\right\}$ and $\left|A\right|\ne 0$. Consider the following two statements;

$\left(P\right)$ If $A\ne I_2$, then $\left|A\right|=-1$

$\left(Q\right)$ If $\left|A\right|=1$, then $tr\left(A\right)=2$

Where $I_2$ denotes $2\times 2$ identity matrix and $tr\left(A\right)$ denotes the sum of the diagonal entries of $A$. Then