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

A real square orthogonal matrix $Q$ satisfiesequivalently $Q^{-1}=Q^T$. Its columns, and also its rows, form an orthonormal basis. Consequently it preserves dot products and Euclidean norms.

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.

Use the order-reversal rule for the transpose of a product:Therefore the product of two orthogonal matrices is orthogonal. The argument does not require $A$ and $B$ to commute.