Idempotent Matrix
Idempotent matrix is a square matrix which when multiplied by itself, gives back the same matrix. A matrix M is said to be an idempotent matrix if M^{2} = M. Further every identity matrix can be termed as an idempotent matrix.
The idempotent matrix is a singular matrix and can have nonzero elements. Let us learn more about the properties of an idempotent matrix with examples, FAQs.
1.  What Is An Idempotent Matrix? 
2.  Properties Of Idempotent Matrix 
3.  Examples On Idempotent Matrix 
4.  Practice Questions 
5.  FAQs on Idempotent Matrix 
What Is An Idempotent Matrix?
Idempotent matrix is a square matrix, which multiplied by itself, gives back the initial square matrix. A matrix M, when multiplied with itself, gives back the same matrix M, M^{2} = M.
Let us consider a matrix A = \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\). Further since A is taken as an idempotent matrix, we have A^{2} = A.
\(\begin{pmatrix}a&b\\c&d\end{pmatrix}\)^{ }× \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) = \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\)
\(\begin{pmatrix}a^2+bc&ab+bd\\ac+cd&bc+d^2\end{pmatrix}\) = \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\)
Here let us compare the terms on either sides.
a^{2} + bc = a
bc = a  a^{2}
ab + bd = b
ab + bd  b = 0
b(a + d  1) = 0
b = 0 or a + d  1 = 0
d = 1  a
From the above derivation we can understand that a matrix A =\(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) is an idempotent matrix if d = 1  a, and bc = a  a^{2}. Further using these two conditions for a 2 x 2 square matrix, we can create an idempotent matrix. Let us create an idempotent matrix by taking a = 5, and we have the other elements of the matrix as follows.
d = 1  a = 1  5 = 4
bc = a  a^{2} = 5  5^{2} = 5  25 = 20
bc = 20
The possible combinations for the values of b and c are b = 10, and c = 2. Hence one of the idempotent matrices which can be formed is as follows.
P = \(\begin{pmatrix}5&10\\2&4\end{pmatrix}\)
Also, all the identity matrices on multiplication with itself give back the identity matrix, and hence the identity matrix is also considered an idempotent matrix.
I_{2 x 2}= \(\begin{bmatrix}1&0\\0&1\end{bmatrix}\)
I_{3x3} = \(\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\)
The determinant of an idempotent matrix is always equal to zero, and hence an idempotent matrix is also a singular matrix.
Properties of Idempotent Matrix
The following are some of the important properties of an idempotent matrix.
 The idempotent matrix is a square matrix.
 The idempotent matrix has an equal number of rows and columns.
 The idempotent matrix is a singular matrix
 The nondiagonal elements can be nonzero elements.
 The eigenvalues of an idempotent matrix is either 0 or 1.
 The trace of an idempotent matrix is equal to the rank of a matrix
 The trace of an idempotent matrix is always an integer.
Related Topics
The following topics help in a better understanding of the idempotent matrix.
Examples on Idempotent Matrix

Example 1: Write an example of a 2 x 2 idempotent matrix.
Solution:
The standard format of an idempotent matrix is A = \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\), and bc = a  a^{2}, d = 1  a
Let us take a = 4, d = 1  4 = 3
bc = 4  4^{2} = 4  16 = 12
Here we can take b = 6, c = 2
A = \(\begin{bmatrix}4&6\\2&3\end{bmatrix}\)
Therefore the idempotent matrix is \(\begin{bmatrix}4&6\\2&3\end{bmatrix}\).

Example 2: Find if the matrix \(\begin{bmatrix}2&2&4\\1&3&4\\1&2&3\end{bmatrix}\) is an idempotent matrix.
Solution:
The given matrix is A = \(\begin{bmatrix}2&2&4\\1&3&4\\1&2&3\end{bmatrix}\).
Let us check this for the idempotent matrix property, A^{2} = A.
A^{2} = \(\begin{bmatrix}2&2&4\\1&3&4\\1&2&3\end{bmatrix}\) × \(\begin{bmatrix}2&2&4\\1&3&4\\1&2&3\end{bmatrix}\)
= \(\begin{bmatrix}2×2+(2)×(1)+(4)×1&2×(2)+(2)×3+(4)×(2)&2×(4)+(2)×4+(4)×(3)\\(1)×2+3×(1)+4×1&(1)×(2)+3×3+4×(2)&(1)×(4)+3×4+4×(3)\\1×2+(2)×(1)+(3)×1&1×(2)+(2)×3+(3)×(2)&1×(4)+(2)×4+(3)×(3)\end{bmatrix}\)
= \(\begin{bmatrix}4+24&46+8&88+12\\23+4&2+98&4+12+412\\2+23&2+66&48+9\end{bmatrix}\)
= \(\begin{bmatrix}2&2&4\\1&3&4\\1&2&3\end{bmatrix}\) = A
Therefore the given matrix is an idempotent matrix.
FAQs on Idempotent Matrix
What Is Idempotent Matrix?
Idempotent matrix is a square matrix which when multiplied by itself, gives back the same matrix. A matrix M is said to be an idempotent matrix if M^{2} = M. Further every identity matrix can be termed as an idempotent matrix.
What Is The Order Of An Idempotent Matrix?
The idempotent matrix has an order of the form n x n. The idempotent matrix is a square matrix with an equal number of rows and columns, and generally, the idempotent matrix is of the order 2 x 2, or 3 x 3.
What Are The Properties Of an Idempotent Matrix?
The three important properties of idempotent matrices are as follows.
 The idempotent matrix is a singular matrix.
 The eigenvalues of an idempotent matrix is either 0 or 1.
 The trace of an idempotent matrix is equal to the rank of a matrix.
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