Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 1 / 18 / a / ii / Solution

Expanding the determinant along the first row givesUnder the matrix trace condition $a^2=3$, this simplifies to $\det M=b(3-b)$. A square matrix is invertible exactly when its determinant is nonzero. Thus