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Involutory Matrix
An involutory matrix is a special type of matrix in mathematics. For a matrix to be involutory, it needs to be an invertible matrix, i.e., a nonsingular square matrix whose inverse exists. An involutory matrix is a square matrix whose product with itself is equal to the identity matrix of the same order. In other words, we can say that an involutory matrix is an inverse of itself. This implies if the square of a matrix is equal to the identity matrix, then it is an involutory matrix.
In this article, we will explore a special type of matrix, named involutory matrix, its definition, and formula. We will also see some examples of the involutory matrix for a better understanding of the concept.
1.  What is Involutory Matrix? 
2.  Involutory Matrix Definition 
3.  Properties of Involutory Matrix 
4.  FAQs on Involutory Matrix 
What is Involutory Matrix?
An involutory matrix is a special kind of matrix as it satisfies the selfinverse function, i.e., an involutory matrix is its own inverse. In simple words, it can be said if the square of a square matrix A of order n is equal to the identity matrix of the same order, then A is an involutory matrix. All involutory matrices of order n are square roots of the identity matrix of order n. Some of the examples of involutory matrices are:
Let us go through the definition of an involutory matrix given below:
Involutory Matrix Definition
A square matrix A of order n × n is said to be an involutory matrix if and only if A^{2} = I, where I is an identity matrix of order n × n. As we know that for two matrices A and B if AB = BA = I, then A and B are inverses of each other. Using the definition of inverse of a matrix, we can say that an involutory matrix is the inverse of itself. The formula for involutory matrix can also be written as, if A is an involutory matrix, then A = A^{1}.
Matrix A = \(\left[\begin{array}{cc}
a & b \\
c & a
\end{array}\right] \) is a general form of a 2 × 2 involutory matrix such that it satisfies a^{2} + bc = 1, where a, b, c are real numbers. Using this we can make involutory matrices such as if we have a = 2, b = 1, and c = 3, then it satisfies the relation a^{2} + bc = 1. So, the matrix \(\left[\begin{array}{cc}
2 & 1 \\
3 & 2
\end{array}\right] \) is an involutory matrix.
Properties of Involutory Matrix
Now that we know the definition of the involutory matrix, let us go through some of its important properties that will help in its application and understanding the concept better:
 If A and B are involutory matrices such that AB = BA, then AB is also an involutory matrix.
 A block diagonal matrix A derived from an involutory matrix is also an involutory matrix.
 Eigenvalues of involutory matrices are always +1 and 1.
 The determinant of any involutory matrix is always ±1.
 Every symmetric involutory matrix is orthogonal and every orthogonal involutory matrix is symmetric.
 If a matrix A is involutory, then A^{n} is also involutory for all integers n. We can say that A^{n} = I if n is even and A^{n} = A if n is odd.
 An involutory matrix A is an idempotent matrix if and only if A is an identity matrix.
Important Notes on Involutory Matrix
 A square matrix A is involutory if and only if A^{2} = I or A = A^{1}.
 All involutory matrices of order n are square roots of the identity matrix of order n.
Related Topics on Involutory Matrix
Involutory Matrix Examples

Example 1: Give an example of an involutory matrix of order 2.
Solution: We know that matrix of the form \(\left[\begin{array}{cc}
a & b \\
c & a
\end{array}\right] \) such it satisfies a^{2} + bc = 1 is an involutory matrix.We can take values a = 3, b = 4, c = 2. It satisfies a^{2} + bc = 1 because 3^{2} + (4)(2) = 9  8 = 1.
Therefore, \(\left[\begin{array}{cc}
3 & 4 \\
2 & 3
\end{array}\right] \) is involutory matrix.Answer: \(\left[\begin{array}{cc}
3 & 4 \\
2 & 3
\end{array}\right] \) 
Example 2: Check if the matrix \(C = \left[\begin{array}{ccc}
5 & 8 & 0 \\
3 & 5 & 0 \\
1 & 2 & 1
\end{array}\right] \)is an involutory matrix.
Solution: To check if matrix C is involutory, we need to determine the value of C^{2}
C^{2} = C.C =
\(C = \left[\begin{array}{ccc}
5 & 8 & 0 \\
3 & 5 & 0 \\
1 & 2 & 1
\end{array}\right] \times \left[\begin{array}{ccc}
5 & 8 & 0 \\
3 & 5 & 0 \\
1 & 2 & 1
\end{array}\right] \\ = \left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right] \)= I
Hence, C is an involutory matrix.
Answer: C is an involutory matrix because C^{2} = I.
FAQs on Involutory Matrix
What is Involutory Matrix in Maths?
An involutory matrix is a special kind of matrix as it satisfies the selfinverse function, i.e., an involutory matrix is its own inverse. An involutory matrix is a square matrix whose product with itself is equal to the identity matrix of the same order.
What is the Involutory Matrix Formula?
The formula for an involutory matrix A is A^{2} = I or A = A^{1}.
What is the Difference Between Idempotent Matrix and Involutory Matrix?
A matrix A is said to be an involutory matrix if A^{2} = I and it is idempotent if A^{2} = A.
How to Check if a Matrix is an Involutory Matrix?
To check if a matrix is involutory, we need to find its product with itself, i.e., A^{2}. If A^{2} = I, where I is an identity matrix, then A is an involutory matrix.
How to Find Involutory Matrix?
Matrix A = \(\left[\begin{array}{cc}
a & b \\
c & a
\end{array}\right] \) is a general form of a 2 × 2 involutory matrix such that it satisfies a^{2} + bc = 1, where a, b, c are real numbers. We can find such values of a, b, c to find involutory matrix.
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