Why is $I$ the only idempotent matrix with nonzero determinant?

Question

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?

Answer 1

Asserting that $A$ is idempotent means that $A^2=A$. But, if $A$ is invertible, then$$\operatorname{Id}=A^{-1}A=A^{-1}A^2=A.$$

Answer 2

$A^2=A \implies A(A-\mathcal{I}_2) = \mathcal{O}_2 $, and since $\text{det}(A)\neq 0$, it follows that $A-\mathcal{I}_2 = \mathcal{O}_2$, i.e. $A$ is the identity matrix.

Answer 3

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.