A matrix is symmetric if it is equal to its transpose. That is, $A = A^T$.

Given $A=\begin{bmatrix}1&4&x\\ z&2&y\\ -3&-1&3\end{bmatrix}$, its transpose is $A^T=\begin{bmatrix}1&z&-3\\ 4&2&-1\\ x&y&3\end{bmatrix}$.

For $A$ to be symmetric, we must have $A = A^T$. Comparing the elements of $A$ and $A^T$, we get:

• $a_{12} = a_{21} \Rightarrow 4 = z$
• $a_{13} = a_{31} \Rightarrow x = -3$
• $a_{23} = a_{32} \Rightarrow y = -1$

Therefore, $x = -3$, $y = -1$, and $z = 4$.

The value of $x+y+z$ is $-3 + (-1) + 4 = 0$.