Step 1: Understand the definition of a Symmetric Matrix

A matrix $A$ is symmetric if and only if $A = A^T$, which implies that the element at row $i$ and column $j$ must be equal to the element at row $j$ and column $i$ for all $i, j$. Mathematically, $a_{ij} = a_{ji}$.

Step 2: Apply the condition to the given matrix

Given $A = \begin{bmatrix} 1 & a & b \\ -1 & 2 & c \\ 0 & 5 & 3 \end{bmatrix}$. Comparing $a_{ij}$ with $a_{ji}$:

For $i=1, j=2$: $a_{12} = a_{21} \implies a = -1$.

For $i=1, j=3$: $a_{13} = a_{31} \implies b = 0$.

For $i=2, j=3$: $a_{23} = a_{32} \implies c = 5$.

Step 3: Calculate the required expression

Substitute the values $a = -1$, $b = 0$, and $c = 5$ into the expression $3a + b + c$: