# Idempotent Matrix

A matrix is defined as a rectangular array of numbers that are arranged in rows and columns. The size of a matrix can be determined by the number of rows and columns in it. A matrix is said to be an “m by n” matrix when it has “m” rows and “n” columns and is written as an “m × n” matrix. For example, a matrix of order “5 × 6” has five rows and six columns. We have different types of matrices like rectangular matrices, square matrices, null matrices, triangular matrices, symmetric matrices, etc.

## Idempotent Matrix Definition

An idempotent matrix is defined as a square matrix that remains unchanged when multiplied by itself. Consider a square matrix “P” of any order, and the matrix P is said to be an idempotent matrix if and only if P^{2} = P. Idempotent matrices are singular and can have non-zero entries. Every identity matrix is also an idempotent matrix, as the identity matrix gives the same matrix when multiplied by itself.

### Examples of Idempotent Matrix

The matrix given below is an idempotent matrix of order “2 × 2.”

The matrix given below is an idempotent matrix of order “3 × 3.”

## Idempotent Matrix Formula

Let us consider a “2 × 2” square matrix . As P is an idempotent matrix, P

^{2}= P.Now, comparing the terms on each side, we get

1) a

^{2}+ bc = a

bc = a − a^{2}2) ab + bd = b

ab + bd − b = 0

b (a + d − 1) = 0

b = 0 or a + d − 1 = 0

d = 1 − aSo, if a matrix is said to be an idempotent matrix, if bc = a − a

^{2}and d = 1 − a.

## Properties of Idempotent Matrix

The following are some important properties of an idempotent matrix:

- Every idempotent matrix is a square matrix.
- All idempotent matrices are singular matrices, apart from the identity matrix.
- The determinant of an idempotent matrix is either one or zero.
- The non-diagonal entries of an idempotent matrix can be non-zero entries.
- The trace of an idempotent matrix is always an integer and equal to the rank of the matrix.
- The eigenvalues of an idempotent matrix are either zero or one.
- The following is the relationship between idempotent and involuntary matrices: A square matrix “A” is said to be an idempotent matrix if and only if P = 2A − I is an involuntary matrix.

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## Solved Examples on Idempotent Matrix

**Example 1: Verify whether the matrix given below is idempotent or not.**

**Solution:**

To prove that the given matrix is idempotent, we have to prove that P

^{2}= P.Hence, verified.

So, the given matrix P is an idempotent matrix.

**Example 2: Verify whether the matrix given below is idempotent or not.**

**Solution:**

To prove that the given matrix is idempotent, we have to prove that B

^{2}= B.Hence, verified.

So, the given matrix B is an idempotent matrix.

**Example 3: Give an example of an idempotent matrix of order 2 × 2.**

**Solution:**

We know that a matrix is said to be an idempotent matrix, if bc = a − a

^{2}and d = 1 − a.Let us consider that a = 5

We have, d = 1 − a

d = 1 − 5 = −4

bc = a − a

^{2}bc = 5 − 25 = −20

Now, let b = 4 and c = −5

So, the matrix is A =

**Example 4: Prove that an identity matrix is an idempotent matrix.**

**Solution:**

To prove that the given matrix is idempotent, we have to prove that I

^{2}= I.Let us consider an identity matrix of order 2 × 2, i.e.,

Hence, proved.

So, an identity matrix is an idempotent matrix.

## FAQs on Idempotent Matrix

**Question 1: What is meant by an idempotent matrix?**

**Answer:**

An idempotent matrix is defined as a square matrix that remains unchanged when multiplied by itself.

**Question 2: How to prove that a matrix is idempotent?**

**Answer:**

Any square matrix “P” is said to be an idempotent matrix if and only if P

^{2}= P. So, to prove that a matrix is idempotent, then the matrix must satisfy the above condition.

**Question 3: Does the inverse of an idempotent matrix exist?**

**Answer:**

We know that the inverse of a square “A” (A

^{-1}) = Adj A/ |A|If the given idempotent matrix is singular, then its inverse does not exist as its determinant is zero.

**Question 4: What is the relationship between an idempotent matrix and an involuntary matrix?**

**Answer:**

The following is the relationship between idempotent and involuntary matrices: A square matrix “A” is said to be an idempotent matrix if and only if

P = 2A − Iis an involuntary matrix.

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