Past exam of the mathematical methods I course of the University of Oxford / 2019 / 11 / c / i / Solution

If $M$ is an $r\times s$ matrix, the product $M^2=MM$ is defined in ordinary matrix multiplication only when $s=r$. Hence the defining equation of an idempotent matrix already requires equal row and column counts:Equivalently, an idempotent linear map is an endomorphism, so the same vector space is both its domain and codomain.