# Idempotent Matrix

Last Updated : 09 Jan, 2023

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 P2 = 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, P2 = P.

Now, comparing the terms on each side, we get

1) a2 + bc = a

bc = a − a2

2) ab + bd = b

ab + bd − b = 0

b (a + d − 1) = 0

b = 0 or a + d − 1 = 0

d = 1 − a

So, if a matrix  is said to be an idempotent matrix, if bc = a − a2 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.

## 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 P2 = 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 B2 = 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 − a2 and d = 1 − a.

Let us consider that a = 5

We have, d = 1 − a

d = 1 − 5 = −4

bc = a − a2

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 I2 = 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?

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?

Any square matrix “P” is said to be an idempotent matrix if and only if P2 = 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?

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.