Linear Algebra

Invertible Idempotent Matrix is the Identity Matrix

Idempotent Matrix Problems and Solutions in Linear Algebra

Problem 1

A square matrix $A$ is called idempotent if $A^2=A$.

Show that a square invertible idempotent matrix is the identity matrix.

Add to solve later

Proof.

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).

The proofs are given in the post ↴
Unit Vectors and Idempotent Matrices

Problem.
Show that

(a) Find a nonzero, nonidentity idempotent matrix.

(b) Show that eigenvalues of an idempotent matrix $A$ is either $0$ or $1$.

See the post ↴
Idempotent Matrix and its Eigenvalues
for solutions.

Add to solve later

More from my site

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 […]
Determine Eigenvalues, Eigenvectors, Diagonalizable From a Partial Information of a Matrix Suppose the following information is known about a $3\times 3$ matrix $A$. \[A\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}=6\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A\begin{bmatrix} 1 \\ -1 \\ 1 […]
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 […]