Question

If $A=\left[\begin{array}{ccc}-1& 2& 3x\\ 2y& 4& -1\\ 6& 5& 0\end{array}\right]$ is a symmetric matrix, then the value of 2x + y is ___________.

Answer 1

A matrix X is a symmetric matrix if X^T = X.

It is given that, the matrix $A=\left[\begin{array}{ccc}-1& 2& 3x\\ 2y& 4& -1\\ 6& 5& 0\end{array}\right]$ is a symmetric matrix.
$\therefore A^T=A$

$\Rightarrow {\left[\begin{array}{ccc}-1& 2& 3x\\ 2y& 4& -1\\ 6& 5& 0\end{array}\right]}^T=\left[\begin{array}{ccc}-1& 2& 3x\\ 2y& 4& -1\\ 6& 5& 0\end{array}\right]$

$\Rightarrow \left[\begin{array}{ccc}-1& 2y& 6\\ 2& 4& 5\\ 3x& -1& 0\end{array}\right]=\left[\begin{array}{ccc}-1& 2& 3x\\ 2y& 4& -1\\ 6& 5& 0\end{array}\right]$

⇒ 3x = 6 and 2y = 2

⇒ x = 2 and y = 1

∴ 2x + y = 2 × 2 + 1 = 5

Thus, the value of 2x + y is 5.

If $A=\left[\begin{array}{ccc}-1& 2& 3x\\ 2y& 4& -1\\ 6& 5& 0\end{array}\right]$ is a symmetric matrix, then the value of 2x + y is _____5_____.