If matrix A=[ array ccc 3 & -2 & 4 1 & 2 & -1 0 & 1 & 1 array ] and...

Given, A=\left[$\begin{array}{ccc}3 & -2 & 4 \\ 1 & 2 & -1 \\ 0 & 1 & 1\end{array}$\right] and $A^{-1}=\frac{1}{k}$\operatorname{adj}(A)$....(i) We know that, A^{-1}=\frac{\operatorname{\text{ adj }}(A)}{|A|}$...(ii) By comparing Eqs. (i) and (iii), $k=|A|$ So,\begin{aligned} |A| & =\left|\begin{array}{ccc} 3 & -2 & 4 \\ 1 & 2 & -1 \\ 0 & 1 & 1 \end{array}\right| \\ & =3(2+1)+2(1+0)+4(1-0)=15 \end{aligned}$$