If $\mathrm{A}=\left[\begin{array}{cc}0& 2\\ 3& -4\end{array}\right]$ and $\mathrm{kA}=\left[\begin{array}{cc}0& 3\mathrm{a}\\ 2\mathrm{b}& 24\end{array}\right]$ then find the value of b - a- k.

We have,

$\begin{array}{l}\mathrm{A}=\left[\begin{array}{cc}0& 2\\ 3& -4\end{array}\right]\Rightarrow \mathrm{kA}=\left[\begin{array}{cc}0& 2\mathrm{k}\\ 3\mathrm{k}& -4\mathrm{k}\end{array}\right]\\ \mathrm{but}, \mathrm{kA}=\left[\begin{array}{cc}0& 3\mathrm{a}\\ 2\mathrm{b}& 24\end{array}\right]\end{array}$

$\begin{array}{l}\therefore \left[\begin{array}{cc}0& 2\mathrm{k}\\ 3\mathrm{k}& -4\mathrm{k}\end{array}\right]\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=\left[\begin{array}{cc}0& 3\mathrm{a}\\ 2\mathrm{b}& 24\end{array}\right]\\ \Rightarrow 2\mathrm{k}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=3\mathrm{a},3\mathrm{k}=2\mathrm{b},-4\mathrm{k}=24\\ \Rightarrow \mathrm{k}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=-6,\mathrm{a}=-4,\mathrm{b}=-9\end{array}$

Hence ,$\mathrm{b}-\mathrm{a}-\mathrm{k}=-9-(-4)-(-6)=-9+4+6=1$