Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 18 / a / ii

Subtracting the second row from the third gives a third row $(0,2,0)$, so expansion along it shows $\det B=-2\ne 0$. Thus the matrix inverse exists and every right-hand side has a unique solution.

To calculate it, temporarily solve $B\mathbf{x}=\mathbf{b}$ with general $\mathbf{b}=(b_1,b_2,b_3{)}^T$. Subtracting the second equation from the third and from the first givesBack-substitution into the second equation givesThereforeFor the given right-hand side, matrix multiplication yields the unique answer