Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 8B / a / Solution

The determinant factors asThus for $\mu \ne \pm 1$, the kernel is $\{0\}$ and the inhomogeneous system has exactly one solution.

At $\mu =-1$, row reduction givesand the augmented matrix has the same rank two as $A$, so there are infinitely many solutions. At $\mu =1$,but the augmented matrix has rank three while $A$ has rank two, so there is no solution. Therefore the number of solutions is