If $A=\left[\begin{array}{ccc}-1 & 2 & 3 x \\ 2 y & 4 & -1 \\ 6 & 5 & 0\end{array}\right]$ is a symmetric matrix, then the value of $2 x+y$ is______
A matrix $X$ is a symmetric matrix if $X^{T}=X$
It is given that, the matrix $A=\left[\begin{array}{ccc}-1 & 2 & 3 x \\ 2 y & 4 & -1 \\ 6 & 5 & 0\end{array}\right]$ is a symmetric matrix.$\therefore A^{T}=A$
$\therefore A^{T}=A$
$\Rightarrow\left[\begin{array}{ccc}-1 & 2 & 3 x \\ 2 y & 4 & -1 \\ 6 & 5 & 0\end{array}\right]^{T}=\left[\begin{array}{ccc}-1 & 2 & 3 x \\ 2 y & 4 & -1 \\ 6 & 5 & 0\end{array}\right]$
$\Rightarrow\left[\begin{array}{ccc}-1 & 2 y & 6 \\ 2 & 4 & 5 \\ 3 x & -1 & 0\end{array}\right]=\left[\begin{array}{ccc}-1 & 2 & 3 x \\ 2 y & 4 & -1 \\ 6 & 5 & 0\end{array}\right]$
$\Rightarrow 3 x=6$ and $2 y=2$
$\Rightarrow x=2$ and $y=1$
$\therefore 2 x+y=2 \times 2+1=5$
Thus, the value of $2 x+y$ is 5 .
If $A=\left[\begin{array}{ccc}-1 & 2 & 3 x \\ 2 y & 4 & -1 \\ 6 & 5 & 0\end{array}\right]$ is a symmetric matrix, then the value of $2 x+y$ is 5