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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.

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3 Answers 3

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It is incorrect, the $0$ matrix is symmetric but not invertable.

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  • $\begingroup$ True! thanks you!! $\endgroup$ Oct 24, 2014 at 2:43
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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.)

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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.

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