1. **Problem:** Find the value of $a$ for which the matrix $$\begin{bmatrix} 2 & 0 & 1 \\ 5 & a & 3 \\ 0 & 3 & 1 \end{bmatrix}$$ is singular. 2. **Formula and rule:** A matrix is singular if its determinant is zero. 3. **Calculate the determinant:** $$\det = 2 \times \begin{vmatrix} a & 3 \\ 3 & 1 \end{vmatrix} - 0 + 1 \times \begin{vmatrix} 5 & a \\ 0 & 3 \end{vmatrix}$$ 4. **Evaluate minors:** $$\begin{vmatrix} a & 3 \\ 3 & 1 \end{vmatrix} = a \times 1 - 3 \times 3 = a - 9$$ $$\begin{vmatrix} 5 & a \\ 0 & 3 \end{vmatrix} = 5 \times 3 - 0 = 15$$ 5. **Substitute back:** $$\det = 2(a - 9) + 15 = 2a - 18 + 15 = 2a - 3$$ 6. **Set determinant to zero for singularity:** $$2a - 3 = 0 \implies 2a = 3 \implies a = \frac{3}{2}$$ **Final answer:** $a = \frac{3}{2}$