Question

A $3\times 3$ matrix is defined as
The values of $x$ required for which $A^{-1}$ can't be determined will be
$A=\left[\begin{array}{ccc}3-x& 2& 2\\ 2& 4-x& 1\\ -2& -4& -1-x\end{array}\right]$

• A. 0, 3, 3
• B. 0, 1, 3
• C. 0, -1, 4
• D. 1, -1, 3

Answer 1

The correct option is A 0, 3, 3

For $A^{-1}$ to be non existent, $|A|=0$
$|A|=\left|\begin{array}{ccc}3-x& 2& 2\\ 2& 4-x& 1\\ -2& -4& -1-x\end{array}\right|=0$
$R_2\to R_2+R_3$
$\left|\begin{array}{ccc}3-x& 2& 2\\ 0& -x& -x\\ -2& -4& -1-x\end{array}\right|=0$
$(-x)\left|\begin{array}{ccc}3-x& 2& 2\\ 0& 1& 1\\ -2& -4& -1-x\end{array}\right|=0$
$R_1\to R_1-2R_2$
$(-x)\left|\begin{array}{ccc}3-x& 0& 0\\ 0& 1& 1\\ -2& -4& -1-x\end{array}\right|=0$
$(-x)(3-x(-1-x+4\left)\right)=0$
$(-x)(3-x)(3-x)=0$
$x=0,3,3$