Find the values of x for which matrix $\begin{bmatrix}3&-1+x&2\\3&-1&x+2\\x+3&-1&2\end{bmatrix}$ is singular.

0, -4

Given matrix is singular

$∴\begin{bmatrix}3&-1+x&2\\3&-1&x+2\\x+3&-1&2\end{bmatrix}=0$

$⇒\begin{bmatrix}0&x&-x\\3&-1&x+2\\x+3&-1&2\end{bmatrix}=0$   $[R_1→R_1-R_2]$

$⇒\begin{bmatrix}0&x&-x\\-x&0&x\\x+3&-1&2\end{bmatrix}=0$   $[R_2→ R_2-R_3]$

$⇒\begin{bmatrix}0&x&0\\-x&0&x\\x+3&-1&1\end{bmatrix}=0$   $[C_3 → C_3 + C_2]$

$⇒-x [(-x)-x(x+3)] = 0$

$⇒x(x^2 + 4x) = 0$

$⇒x = 0, -4$