If $A=\left[\begin{array}{cc}4 & x+2 \\ 2 x-3 & x+1\end{array}\right]$ is a symmetric matrix, then $x=$_______
The given matrix $A=\left[\begin{array}{cc}4 & x+2 \\ 2 x-3 & x+1\end{array}\right]$ is symmetric.
$\therefore A^{T}=A$
$\Rightarrow\left[\begin{array}{cc}4 & x+2 \\ 2 x-3 & x+1\end{array}\right]^{T}=\left[\begin{array}{cc}4 & x+2 \\ 2 x-3 & x+1\end{array}\right]$
$\Rightarrow\left[\begin{array}{cc}4 & 2 x-3 \\ x+2 & x+1\end{array}\right]=\left[\begin{array}{cc}4 & x+2 \\ 2 x-3 & x+1\end{array}\right]$
$\Rightarrow 2 x-3=x+2$
$\Rightarrow x=5$
If $A=\left[\begin{array}{cc}4 & x+2 \\ 2 x-3 & x+1\end{array}\right]$ is a symmetric matrix, then $x=\underline{5}$
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