Let $A$ be an $n \times n$ invertible idempotent matrix.

Since $A$ is invertible, the inverse matrix $A^{-1}$ of $A$ exists and it satisfies $A^{-1} A=I_n$, where $I_n$ is the $n\times n$ identity matrix.

Since $A$ is idempotent, we have $A^2=A$.
Multiplying this equality by $A^{-1}$ from the left, we get $A^{-1}A^2=A^{-1}A$. Using the fact that $A^{-1} A=I_n$, we obtain $A=I_n$.

The proof is completed.

Related Question.

Give it a try with the following problems about idempotent matrices.

Problem. (a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$.
Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$.

Prove that $P$ is an idempotent matrix.

(b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be unit vectors in $\R^n$ such that $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.
Let $Q=\mathbf{u}\mathbf{u}^{\trans}+\mathbf{v}\mathbf{v}^{\trans}$.

Prove that $Q$ is an idempotent matrix.

(c) Prove that each nonzero vector of the form $a\mathbf{u}+b\mathbf{v}$ for some $a, b\in \R$ is an eigenvector corresponding to the eigenvalue $1$ for the matrix $Q$ in part (b).

Idempotent Matrices. 2007 University of Tokyo Entrance Exam Problem
For a real number $a$, consider $2\times 2$ matrices $A, P, Q$ satisfying the following five conditions.
$A=aP+(a+1)Q$
$P^2=P$
$Q^2=Q$
$PQ=O$
$QP=O$,
where $O$ is the $2\times 2$ zero matrix.
Then do the following problems.
(a) Prove that […]

Unit Vectors and Idempotent Matrices
A square matrix $A$ is called idempotent if $A^2=A$.
(a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$.
Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$.
Prove that $P$ is an idempotent matrix.
(b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be […]

The Inverse Matrix of the Transpose is the Transpose of the Inverse Matrix
Let $A$ be an $n\times n$ invertible matrix. Then prove the transpose $A^{\trans}$ is also invertible and that the inverse matrix of the transpose $A^{\trans}$ is the transpose of the inverse matrix $A^{-1}$.
Namely, show […]

Using Properties of Inverse Matrices, Simplify the Expression
Let $A, B, C$ be $n\times n$ invertible matrices. When you simplify the expression
\[C^{-1}(AB^{-1})^{-1}(CA^{-1})^{-1}C^2,\]
which matrix do you get?
(a) $A$
(b) $C^{-1}A^{-1}BC^{-1}AC^2$
(c) $B$
(d) $C^2$
(e) $C^{-1}BC$
(f) $C$
Solution.
In this problem, we […]

A One Side Inverse Matrix is the Inverse Matrix: If $AB=I$, then $BA=I$
An $n\times n$ matrix $A$ is said to be invertible if there exists an $n\times n$ matrix $B$ such that
$AB=I$, and
$BA=I$,
where $I$ is the $n\times n$ identity matrix.
If such a matrix $B$ exists, then it is known to be unique and called the inverse matrix of $A$, denoted […]

Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite
Suppose $A$ is a positive definite symmetric $n\times n$ matrix.
(a) Prove that $A$ is invertible.
(b) Prove that $A^{-1}$ is symmetric.
(c) Prove that $A^{-1}$ is positive-definite.
(MIT, Linear Algebra Exam Problem)
Proof.
(a) Prove that $A$ is […]

Sherman-Woodbery Formula for the Inverse Matrix
Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\R^n$, and let $I$ be the $n \times n$ identity matrix. Suppose that the inner product of $\mathbf{u}$ and $\mathbf{v}$ satisfies
\[\mathbf{v}^{\trans}\mathbf{u}\neq -1.\]
Define the matrix […]