If the matrix begin{bmatrix} x & x^2+3x & 5 | vedclass

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(A) For a symmetric matrix, $A^T = A$, which implies $A_{ij} = A_{ji}$ for all $i, j$.Equating the corresponding elements:$A_{12} = A_{21} \Rightarrow

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-$2$

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(A) For a symmetric matrix, $A^T = A$, which implies $A_{ij} = A_{ji}$ for all $i, j$.Equating the corresponding elements:$A_{12} = A_{21} \Rightarrow x^2 + 3x = -2x - 6$$x^2 + 5x + 6 = 0$$(x + 2)(x + 3) = 0 \Rightarrow x = -2$ or $x = -3$.$A_{23} = A_{32} \Rightarrow -4x - 2 = x^2 + 2$$x^2 + 4x + 4 = 0$$(x + 2)^2 = 0 \Rightarrow x = -2$.Since both conditions must be satisfied simultaneously, the common value is $x = -2$.

If the matrix $\begin{bmatrix} x & x^2+3x & 5 \\ -2x-6 & x^2 & -4x-2 \\ 5 & x^2+2 & x^3 \end{bmatrix}$ is a symmetric matrix, then the value of $x$ is

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