Why is $I$ the only idempotent matrix with nonzero determinant? While reading about the attribute of the identity Matrix, it's mentioned that I is not only idempotent but that it is also the only such matrix that does not have a determinant of zero. While I being idempotent is simple to understand, how is it proved that every other matrix without a determinant of zero isn't?

Idempotent means
$$M^2 = M$$ Which means $$(M-I) M = 0.$$ So, if the determinant of $M$ is not $0,$ $M-I$ must be singular, so there must be a vector $v$ such that $Mv = v.$ Now, consider the orthogonal complement of $v.$ The matrix is still idempotent on that, so do it by induction.
