"Let P = $$\left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} & 0 \cr } } \right]$$, where $\alpha$ $\in$ R. Suppose Q = [ qij] is a matrix satisfying PQ = kl3 for some non-zero k $\in$ R. If q23 = $- {k \over 8}$ and |Q| = ${{{k^2}} \over 2}$, then a2 + k2 is equal to ______."

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"As $PQ = kI \Rightarrow Q = k{P^{ - 1}}I$

now $Q = {k \over {|P|}}(adjP)I$

$$\Rightarrow Q = {k \over {(20 + 12\alpha )}}\left[ {\matrix{ - & - & - \cr - & - & {( - 3\alpha - 4)} \cr - & - & - \cr } } \right]\left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right]$$

$\because$ ${q_{23}} = {{ - k} \over 8}$

$\Rightarrow {k \over {(20 + 12\alpha )}}( - 3\alpha - 4) = {{ - k} \over 8}$

$\Rightarrow 2(3\alpha + 4) = 5 + 3\alpha$

$3\alpha = - 3 \Rightarrow \alpha = - 1$

also $$|Q| = {{{k^3}|I|} \over {|P|}} \Rightarrow {{{k^2}} \over 2} = {{{k^3}} \over {(20 + 12\alpha )}}$$

$\Rightarrow$ $(20 + 12\alpha ) = 2k \Rightarrow 8 = 2k \Rightarrow k = 4$"

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Let P = $$\left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} & 0 \cr } } \right]$$, where $\alpha$ $\in$ R. Suppose Q = [ qij] is a matrix satisfying PQ = kl3 for some non-zero k $\in$ R. If q23 = $- {k \over 8}$ and |Q| = ${{{k^2}} \over 2}$, then a2 + k2 is equal to ______.

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