1. **State the problem:** We are given a matrix $\begin{bmatrix} x & 3 & x \\ x & x-2 & ? \end{bmatrix}$ (assuming the matrix is square and the last element is missing or a typo, let's consider a 2x2 matrix $\begin{bmatrix} x & 3 \\ x & x-2 \end{bmatrix}$) and asked to find the values of $x$ for which this matrix is singular. 2. **Recall the definition of a singular matrix:** A matrix is singular if its determinant is zero. 3. **Write the determinant formula for a 2x2 matrix:** $$\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$$ 4. **Apply the formula to our matrix:** $$\det = x(x-2) - 3x = x^2 - 2x - 3x = x^2 - 5x$$ 5. **Set the determinant equal to zero to find singular values:** $$x^2 - 5x = 0$$ 6. **Factor the equation:** $$x(x - 5) = 0$$ 7. **Solve for $x$:** $$x = 0 \quad \text{or} \quad x = 5$$ **Final answer:** The matrix is singular when $x = 0$ or $x = 5$.