Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 1 / d / Solution

The matrix is invertible precisely when $a\ne 0$, andThe largest group is thereforethe real affine group $\mathbb{R}⋊{\mathbb{R}}^{\ast }$ under $(a,b)(a^{\prime },b^{\prime })=(a{a}^{\prime },b+a{b}^{\prime })$.