Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof resit / 4 / c

If $A$ is $m\times n$, then $Y$ is $n\times m$. The equation $AY=I_m$ makes every standard basis vector of ${\mathbb{R}}^m$ a linear combination of the columns of $A$: take the corresponding column of $Y$ as coefficients. Hence the column space is all of ${\mathbb{R}}^m$, giving $\mathrm{rank}(A)=m$. In particular, $n\ge m$ and $A$ has full row rank.

The system is consistent for every $b\in {\mathbb{R}}^m$: take $x=Yb$, since $A(Yb)=(AY)b=b$. Thus $Y$ is a right inverse giving an explicit solution for every right-hand side.

All solutions are $Yb+u$ with $u$ in the kernel of a linear map represented by $A$. The rank-nullity theorem gives kernel dimension $n-m$; the solution is unique when $n=m$, and is nonunique when $n>m$.