Question

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

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

Answer 1

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

A matrix is singular if its determinant is $0$.

$\left|\begin{array}{ccc}3-x& 2& 2\\ 2& 4-x& 1\\ -2& -4& -1-x\end{array}\right|=0$

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

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

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

$=3x^2-9x-x^3+3x^2+4x-4x=0$

$=6x^2-9x-x^3=0$

$=x^3+9x-6x^2=0$

$=x(x^2-6x+9)=0$

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

$\therefore x=0,3$.