Question

For what value of $x$, the matrix $A$ is singular?$A=\left[\begin{array}{ccc}3-x& 2& 2\\ 2& 4-x& 1\\ -2& -4& -1-x\end{array}\right]$

• A. $x=0,2$
• B. $x=1,2$
• C. $x=2,3$
• D. $x=0,3$

Answer 1

The correct option is D $x=0,3$

$A=\left[\begin{array}{ccc}3-x& 2& 2\\ 2& 4-x& 1\\ -2& -4& -1-x\end{array}\right]$

If the matrix is singular, its determinant has to be zero.

$\Rightarrow (3-x)\left[\right(4-x\left)\right(-1-x)+4]-2\left[2\right(-1-x)+2]+2[-8+2(4-x\left)\right]=0$

$\Rightarrow (3-x)[-4-4x+x+x^2+4]-2[-2-2x+2]+2[-8+8-2x]=0$

$\Rightarrow (3-x)[x^2-3x]+4x-4x=0$

$\Rightarrow (3-x)x(x-3)=0$

$\Rightarrow x=0,3$