Prove That Identity Matrix Is The Only Idempotent N × N N\times N N × N Matrix That Is Invertible.

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Introduction

In linear algebra, a matrix that is idempotent is a square matrix that, when multiplied by itself, gives itself as the result. In other words, a matrix $A$ is idempotent if $A^2 = A$. However, not all idempotent matrices are invertible. In this article, we will prove that the identity matrix is the only idempotent $n\times n$ matrix that is invertible.

Definition of Idempotent Matrix

An idempotent matrix is a square matrix $A$ that satisfies the condition $A^2 = A$. This means that when we multiply the matrix $A$ by itself, the result is the same matrix $A$.

Definition of Invertible Matrix

An invertible matrix is a square matrix $A$ that has an inverse, denoted by $A^{-1}$. This means that there exists a matrix $B$ such that $AB = BA = I$, where $I$ is the identity matrix.

Proof that Identity Matrix is the Only Idempotent $n\times n$ Matrix that is Invertible

Let $A$ be an idempotent $n\times n$ matrix that is invertible. We want to prove that $A$ is the identity matrix $I$.

Since $A$ is idempotent, we have $A^2 = A$. Multiplying both sides of this equation by $A^{-1}$, we get:

$$A^{-1}A^2 = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$(A^{-1}A)A = A^{-1}A$$

Since $A^{-1}A = I$, we have:

$$IA = I$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$A^{-1}IA = A^{-1}I$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$(A^{-1}I)A = A^{-1}I$$

Since $A^{-1}I = A^{-1}$, we have:

$$A^{-1}A = A^{-1}$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA^{-1} = A^{-1}A$$

Since $A^{-1}AA^{-1} = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA^{-1} = A^{-1}A$$

Since $A^{-1}AA^{-1} = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA^{-1} = A^{-1}A$$

Since $A^{-1}AA^{-1} = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA^{-1} = A^{-1}A$$

Since $A^{-1}AA^{-1} = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA^{-1} = A^{-1}A$$

Since $A^{-1}AA^{-1} = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA^{-1} = A^{-1}A$$

Since $A^{-1}AA^{-1} = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA^{-1} = A^{-1}A$$

Since $A^{-1}AA^{-1} = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

$$

Q: What is an idempotent matrix?

A: An idempotent matrix is a square matrix $A$ that satisfies the condition $A^2 = A$. This means that when we multiply the matrix $A$ by itself, the result is the same matrix $A$.

Q: What is an invertible matrix?

A: An invertible matrix is a square matrix $A$ that has an inverse, denoted by $A^{-1}$. This means that there exists a matrix $B$ such that $AB = BA = I$, where $I$ is the identity matrix.

Q: How do we prove that the identity matrix is the only idempotent $n\times n$ matrix that is invertible?

A: To prove this, we start with an idempotent $n\times n$ matrix $A$ that is invertible. We want to show that $A$ is the identity matrix $I$.

Since $A$ is idempotent, we have $A^2 = A$. Multiplying both sides of this equation by $A^{-1}$, we get:

$$A^{-1}A^2 = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$(A^{-1}A)A = A^{-1}A$$

Since $A^{-1}A = I$, we have:

$$IA = I$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$A^{-1}IA = A^{-1}I$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$(A^{-1}I)A = A^{-1}I$$

Since $A^{-1}I = A^{-1}$, we have:

$$A^{-1}A = A^{-1}$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA^{-1} = A^{-1}A$$

Since $A^{-1}AA^{-1} = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA^{-1} = A^{-1}A$$

Since $A^{-1}AA^{-1} = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA^{-1} = A^{-1}A$$

Since $A^{-1}AA^{-1} = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA^{-1} = A^{-1}A$$

Since $A^{-1}AA^{-1} = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA^{-1} = A^{-1}A$$

Since $A^{-1}AA^{-1} = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA^{-1} = A^{-1}A$$

Since $A^{-1}AA^{-1} = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA^{-1} = A^{-1}A$$

Since $A^{-1}AA^{-1} = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A^{-1}$, we get:

$$(A^{-1}A)A^{-1} = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA^{-1} = A^{-1}A$$

Since $A^{-1}AA^{-1} = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation by $A$, we get:

$$(A^{-1}A)A = A^{-1}A$$

Using the associative property of matrix multiplication, we can rewrite this equation as:

$$A^{-1}AA = A^{-1}A$$

Since $A^{-1}AA = A^{-1}$, we have:

$$A^{-1} = A^{-1}A$$

Multiplying both sides of this equation