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Idempotent matrix is a matrix when multiplied by itself will produce itself (same matrix). It is an idempotent matrix if A x A = A, where A is an n x n matrix.

## The Definition of Idempotent Matrix

Just as stated above, matrix A is an idempotent matrix if we multiply it with itself and produces the very same matrix, or in n x n matrix:

Thus,

Why is it known as “idempotent”?

It is called idempotent because of its property:

Idempotence is a property when a computer or mathematical operation is executed or multiplied several times and it still produces the same result.

## Example

Idempotent matrix comes with several forms, it can be 2 x 2 matrix, 3 x 3 matrix, 4 x 4 matrix, and so on.

For a 2 x 2 matrix:

For a 3 x 3 matrix:

## How to Check if it is an Idempotent Matrix?

Well, from the example above, you should have understood how to prove whether a matrix is an idempotent matrix or not.

Say, you have Matrix A (doesn’t matter whether it is 2×2, 3×3, or more).

Next, just multiply Matrix A with Matrix A.

If the result is exactly the same as Matrix A, we can conclude that Matrix A is an idempotent matrix.

Hence,

Otherwise, if the result is not exactly the same as matrix A, we can conclude that Matrix A is not an idempotent matrix.

It is quite similar to a nilpotent matrix.

## Formula

The formula we need to verify an idempotent matrix can be seen above, now we will try to make an idempotent matrix.

Assume we want to make a 2×2 idempotent matrix

If we want to make it, we can put any number for a-b-c-d but we have to make sure that:

As long as this rule is fulfilled, other numbers don’t matter.

Or we can use

Using a simple 2×2 matrix, if the matrix above is idempotent, then

**Properties of Idempotent Matrix**

What are the important things to remember about an idempotent matrix? There are several properties of an idempotent matrix, such as:

- Identity matrix is an idempotent matrix.
- An idempotent matrix is a nonsingular matrix.
- For a non-identity idempotent matrix, its independent rows (and columns) number is less than its rows (and columns) number.
- If an idempotent matrix is subtracted from an identity matrix, the result is also idempotent.
- The eigenvalues of an idempotent matrix is always 0 or 1.
- The determinant of an idempotent matrix is always 0 or 1.
- An idempotent matrix is always diagonalizable.
- The trace of an idempotent matrix (the sum of the values on its main diagonal) is equal to its rank of the matrix and always an integer.

## Frequently Asked Questions

**How do you check if a matrix is Idempotent or not?**

Idempotent matrix is a matrix when multiplied by itself will produce itself (same matrix). It is an idempotent matrix if A x A = A, where A is an n x n matrix.

**What is an example of an idempotent matrix?**

The easiest example of an idempotent matrix is an identity matrix where it only contains 1 on its main diagonal and 0 for the rest.

**What is an idempotent matrix formula?**

An idempotent matrix is a square matrix that we multiply with itself and produces the very same matrix. A matrix is said idempotent only if A x A = A.

**Is an idempotent matrix always 0 or 1?**

The determinant of an idempotent matrix is always either zero or one.