Is there an easy way to invert a 3x3 symmetric matrix?
for example A = $\begin{pmatrix} -1& 2& 0\\ 2& -5& 0\\ 0& 0& -1\\ \end{pmatrix}$
is there any 'short-cut' method? I know I can use gaussian elimination, but that is quite long, so if theres an easier way, can someone please show me ? thank you.
It's important to note that in general symmetric matrices are not invertible. For instance:
$$ \left [ \begin{array}{cc} 1 & 0 \\ 0 & 0 \\ \end{array} \right ]. $$
Given however that a symmetric matrix $A$ is invertible, then by the spectral theorem we are guaranteed an orthonormal basis of eigenvectors which means we can give $A$ the diagonalized form $A = PDP^T$ where $D$ is the diagonal matrix of eigenvalues, and $P$ is the orthogonal matrix whose columns are the corresponding eigenvectors (note that for orthogonal matrices that $P^{-1} = P^T$). To find the inverse of $A$ (provided all eigenvalues are nonzero) we simply need to find the inverse of $D$. The reason why is that if $D = diag(\lambda_1, \ldots, \lambda_n)$ then $D^{-1} = diag \left (\frac{1}{\lambda_1}, \ldots, \frac{1}{\lambda_n} \right )$. We can then write $A^{-1} = PD^{-1}P^T$ since
$$ A^{-1}A \;\; =\;\; (PD^{-1}P^T)(PDP^T) \;\; =\;\; PD^{-1}DP^T \;\; =\;\; PP^T \;\; = \;\; I. $$