# If $A$ is an Idempotent Matrix, then When $I-kA$ is an Idempotent Matrix?

## Problem 426

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

**(a)** Suppose $A$ is an $n \times n$ idempotent matrix and let $I$ be the $n\times n$ identity matrix. Prove that the matrix $I-A$ is an idempotent matrix.

**(b)** Assume that $A$ is an $n\times n$ nonzero idempotent matrix. Then determine all integers $k$ such that the matrix $I-kA$ is idempotent.

**(c)** Let $A$ and $B$ be $n\times n$ matrices satisfying

\[AB=A \text{ and } BA=B.\]
Then prove that $A$ is an idempotent matrix.

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## Proof.

### (a) Prove that the matrix $I-A$ is an idempotent matrix.

To prove that $I-A$ is an idempotent matrix, we show that $(I-A)^2=I-A$.

We compute

\begin{align*}

(I-A)^2&=(I-A)(I-A)\\

&=I(I-A)-A(I-A)\\

&=I-A-A+A^2\\

&=I-2A+A^2\\

&=I-2A+A && \text{since $A$ is idempotent}\\

&=I-A.

\end{align*}

Thus, we have $(I-A)^2=I-A$ and $I-A$ is an idempotent matrix.

### (b) Determine all integers $k$ such that the matrix $I-kA$ is idempotent.

Let us find out the condition on $k$ so that $I-kA$ is an idempotent matrix.

We have

\begin{align*}

&(I-kA)^2=(I-kA)(I-kA)\\

&=I(I-kA)-kA(I-kA)\\

&=I-kA-kA+k^2A^2\\

&=I-2kA+k^2A && \text{ since $A$ is idempotent}\\

&=I-(2k-k^2)A.

\end{align*}

It follows that $I-kA$ is idempotent if and only if $I-kA=I-(2k-k^2)A$, or equivalently $(k^2-k)A=O$, the zero matrix.

Since $A$ is not the zero matrix, we see that $I-kI$ is idempotent if and only if $k^2-k=0$.

Since $k^2-k=k(k-1)$, we conclude that $I-kA$ is an idempotent matrix if and only if $k=0, 1$.

### (c) Prove that $A$ is an idempotent matrix.

Let $A$ and $B$ be $n\times n$ matrices satisfying

\[AB=A \tag{*}\]
and

\[ BA=B. \tag{**}\]
Our goal is to show that $A^2=A$.

We compute

\begin{align*}

&A^2=AA\\

&=(AB)A && \text{by (*)}\\

&=A(BA)\\

&=AB && \text{by (**)}\\

&=A && \text{by (*)}.

\end{align*}

Therefore, we have obtained the identity $A^2=A$, and we conclude that $A$ is an idempotent matrix.

## Related Question.

**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

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