If $A = \begin{bmatrix} -2 & -1 & 1 \\ -1 & 7 & 4 \\ 1 & -x & -3 \end{bmatrix}$ be symmetric matrix then find the value of $x$.
Step-by-Step Solution
Key Concept: A symmetric matrix is defined by the property $A = A^T$, which implies that elements across the main diagonal are equal ($a_{ij} = a_{ji}$).
<div>$x = -4$<br>for symmetric matrix $A = A^T$<br>$\Rightarrow \begin{bmatrix} -2 & -1 & 1 \\ -1 & 7 & 4 \\ 1 & -x & -3 \end{bmatrix} = \begin{bmatrix} -2 & -1 & 1 \\ -1 & 7 & -x \\ 1 & 4 & -3 \end{bmatrix}$<br>$\therefore x = -4$</div>
Correct Answer: -4