Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 13 / b

Write the linear system as $Ar=d$ withExpansion along the first row gives $\det A=(16-15)-2(12-5)+3(9-4)=2$. Since $\det A=2\ne 0$, $A$ is an invertible matrix, and $r=A^{-1}d$ is the unique solution for this right side.

The adjugate matrix formula givesMultiplication by the right-side column yieldsMultiplying the inverse by $A$ checks the identity matrix, and substitution of this solution into the three original rows gives $1,2,3$ respectively.