If \( A=\begin{bmatrix}3 & -2 & 3\\2 & 1 & -1\\4 & -3 & 2\end{bmatrix} \) and \( A(\mathrm{adj}\,A)=xI \) , where \( I \) is an identity matrix of order 3, then \( x \) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → -17
For every square matrix, \( A(\mathrm{adj}\,A)=|A|\,I \), so x = |A|. Expanding along the first row:
\( \displaystyle |A|=3\begin{vmatrix}1 & -1\\-3 & 2\end{vmatrix}-(-2)\begin{vmatrix}2 & -1\\4 & 2\end{vmatrix}+3\begin{vmatrix}2 & 1\\4 & -3\end{vmatrix} \)
\( \displaystyle =3(2-3)+2(4+4)+3(-6-4) \)
\( \displaystyle =-3+16-30 \)
\( \displaystyle =-17 \)
So x = −17.