Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 2 / 12 / a / ii / Solution

To invert the matrix, row-reduce the augmented matrix $(M\mid I)$. Subtracting row one from rows two and three gives left rows $(0,-4,1)$ and $(0,0,2)$, so the pivots are nonzero and back-substitution is possible. The resulting matrix inverse isMultiplying by $M$ verifies the identity matrix. This inverse is not $M^T$; indeedThus $M$ is not orthogonal.