Is any symmetric matrix invertible? I'm trying to prove this theoretical question, but I don't know what I need to do. I apologize for the simple question, but I'm in doubt and need clarification.
$\endgroup$3 Answers
$\begingroup$It is incorrect, the $0$ matrix is symmetric but not invertable.
$\endgroup$ 1 $\begingroup$Adding a couple of non zero example for future reference: $$\left[\begin{matrix} 1 & 1 \\ 1 & 1\end{matrix}\right]$$ is symmetric; but not invertible.
Also, $$\left[\begin{matrix} 2 & 2 & 1 \\ 2 & 2 & 1 \\ 1 & 1 & 1\end{matrix}\right]$$ $$\left[\begin{matrix} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 1\end{matrix}\right]$$ and the list goes on. (they are all singular, that is, determinant is zero.)
$\endgroup$ $\begingroup$Since others have already shown that not all symmetric matrices are invertible, I will add when a symmetric matrix is invertible.
A symmetric matrix is positive-definite if and only if its eigenvalues are all positive. The determinant is the product of the eigenvalues. A square matrix is invertible if and only if its determinant is not zero. Thus, we can say that a positive definite symmetric matrix is invertible.
$\endgroup$
