1. **Problem Statement:** Check whether the given matrix $$\begin{bmatrix}3 & 0 & 1 \\ 2 & 0 & 4 \\ 1 & 5 & 2\end{bmatrix}$$ is symmetric, diagonal, or singular. 2. **Definitions and Formulas:** - A matrix $A$ is **symmetric** if $A = A^T$, where $A^T$ is the transpose of $A$. - A matrix is **diagonal** if all off-diagonal elements are zero. - A matrix is **singular** if its determinant $|A| = 0$. 3. **Check Symmetry:** Calculate transpose $A^T$: $$A^T = \begin{bmatrix}3 & 2 & 1 \\ 0 & 0 & 5 \\ 1 & 4 & 2\end{bmatrix}$$ Compare with $A$: $$A \neq A^T$$ So, matrix is **not symmetric**. 4. **Check Diagonal:** Matrix $A$ has non-zero off-diagonal elements (e.g., $2$, $1$, $5$, $4$), so it is **not diagonal**. 5. **Check Singularity:** Calculate determinant $|A|$: $$|A| = 3 \times \begin{vmatrix}0 & 4 \\ 5 & 2\end{vmatrix} - 0 \times \begin{vmatrix}2 & 4 \\ 1 & 2\end{vmatrix} + 1 \times \begin{vmatrix}2 & 0 \\ 1 & 5\end{vmatrix}$$ Calculate minors: $$\begin{vmatrix}0 & 4 \\ 5 & 2\end{vmatrix} = (0)(2) - (4)(5) = -20$$ $$\begin{vmatrix}2 & 0 \\ 1 & 5\end{vmatrix} = (2)(5) - (0)(1) = 10$$ So, $$|A| = 3(-20) + 0 + 1(10) = -60 + 10 = -50$$ Since $|A| \neq 0$, matrix is **non-singular**. **Final answer:** The matrix is neither symmetric nor diagonal, but it is non-singular.