Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 1 / 18 / a

Expanding the determinant along the first row gives

The matrix of cofactors is $\left(\begin{array}{ccc}-1& 2& -3\\ 2& 0& -2\\ -3& -2& 3\end{array}\right)$. Transposing and dividing by the nonzero determinant gives the inverse matrixMultiplication by $S$ verifies the identity matrix.

Take the diagonal matrix $T=\mathrm{diag}(1,0,0)$. Multiplying on the left retains the first row of $S$, while multiplying on the right retains its first column. ThusTheir $(1,3)$ entries differ, so this is an explicit example of noncommuting matrices.