Q79 / 120 · VITEEE 2017 · Mathematics
If matrix $A=\left[\begin{array}{ccc}3 & -2 & 4 \\ 1 & 2 & -1 \\ 0 & 1 & 1\end{array}\right]$ and $\mathrm{A}^{-1}=\frac{1}{\mathrm{k}} \operatorname{adj}(\mathrm{A})$, then $k$ is
- A7
- B$-7$
- Correct answerC15
- D$-11$
Solution
If $\mathrm{A}=\left[\begin{array}{ccc}3 & -2 & 4 \\ 1 & 2 & -1 \\ 0 & 1 & 1\end{array}\right]$ and $\mathrm{A}^{-1}=\frac{1}{k} \operatorname{adj}(\mathrm{A})$......(i)
Also, we know $\mathrm{A}^{-1}=\frac{\operatorname{adj}(\mathrm{A})}{|\mathrm{A}|}$......(ii)
$\therefore$ By comparing (i) and (ii)
$|\mathrm{A}|=k$
$$
\begin{array}{l}
\Rightarrow|\mathrm{A}|=\left|\begin{array}{ccc}
3 & -2 & 4 \\
1 & 2 & -1 \\
0 & 1 & 1
\end{array}\right| \\
=3(2+1)+2(1+0)+4(1-0) \\
=9+2+4=15
\end{array}
$$
Hence, the correct answer is option: 'C'.