# Involutory Matrix

Involutory Matrix is defined as the matrix that follows self inverse function i.e. the inverse of the Involutory matrix is the matrix itself. 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 Ã— n” matrix. For example, a matrix of order “5 Ã— 6” has five rows and six columns. We have different types of matrices, like rectangular, square, triangular, symmetric, singular, etc.

## Involutory Matrix

An involutory matrix is a special type of matrix whose square is equal to an identity matrix. Only square and invertible matrices can be Involutory Matrices. A square matrix is said to be an involutory matrix that, when multiplied by itself, gives an identity matrix of the same order. A square matrix “P” is said to be an involutory matrix if its inverse is the original matrix itself i.e. P = P-1.

### Examples of involutory Matrix

• The matrix given below is an involutory matrix of order “2 Ã— 2.”

• The matrix given below is an involutory matrix of order “3 Ã— 3.”

## involutory Matrix Formula

Let us consider a “2 Ã— 2” square matrix . The given matrix is said to be an involutory matrix if satisfies the condition A2 = I

Now, comparing the terms on each side, we get

a2 + bc = 1

ab + bd = 0

b (a + d) = 0

b = 0 or a + d = 0

d = âˆ’a

So, a square matrix  is said to be an involutory matrix if

• a2 + bc = 1
• d = âˆ’a

## Properties of involutory Matrix

The following are some important properties of an involutory matrix:

• A square matrix “A” of any order is said to be involutory if and only if A2 = I or A = A-1.
• If A and B are two involutory matrices of the same order and AB = BA, then AB is also an involutory matrix.
• The determinant of an involutory matrix is always either -1 or +1.
• If “A” is an involutory matrix of any order, then An = I if n is even and An = A if n is odd, where n is an integer.
• If a block diagonal matrix is derived from an involutory matrix, then the obtained matrix is also involutory.
• The eigenvalues of an involutory matrix are always either -1 or +1.
• Symmetric involutory matrix is orthogonal, and vice versa.
• An involutory matrix “A” can also be an idempotent matrix if “A” is an identity matrix.
• The following is the relationship between idempotent and involutory matrices: A square matrix “A” is said to be an involutory matrix if and only if A = Â½ (B + I), where B is an idempotent matrix.

## Solved Examples on involutory Matrix

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

Solution:

To prove that the given matrix is involutory, we have to prove that A2 = A.

Hence, verified.

So, the given matrix A is an involutory matrix.

Example 2: Give an example of an involutory matrix of order 2 Ã— 2.

Solution:

A matrix  is said to be an involutory matrix, if a2 + bc = 1.

Let us consider that a = 3, b = 4, c = âˆ’2 such that a2 + bc = 1.

(3)2 + (4) Ã— (âˆ’2) = 9 âˆ’ 8 = 1

We know that d = âˆ’a.

So, the involutory matrix is .

Example 3: Prove that the matrix given below is involutory.

Solution:

To prove that the given matrix is involutory, we have to prove that B = B-1.

|B| = âˆ’49 âˆ’ (âˆ’48) = âˆ’1

Hence, the given matrix is involutory.

Example 4: Prove that the determinant of an involutory matrix given below is always Â±1.

Solution:

Let us consider of an involutory matrix “P” of order “n Ã— n” to prove that its determinant is always Â±1.

We know that a square matrix “P” is said to be involutory if and only if P2 = I.

P Ã— P = I

Now, |P| Ã— |P| = |I|

We know that the determinant of an identity matrix of any order is 1.

(|P|)2 = 1

|P| = âˆš1 = Â±1

Thus, the determinant of an involutory matrix of any order is always Â±1.

Hence proved.

## FAQs on involutory Matrix

### Question 1: How to prove that a matrix is involutory?

Any square matrix “P” is said to be an involutory matrix if and only if P2 = I or P = P-1. So, to prove that a matrix is involutory, the matrix must satisfy the above condition.

### Question 2: Define an involutory matrix.

Solution:

A square matrix is said to be an involutory matrix that, when multiplied by itself, gives an identity matrix of the same order.

### Question 3: What is the relation between involutory and idempotent matrices?

Solution:

The following is the relationship between idempotent and involutory matrices: A square matrix “A” is said to be an involutory matrix if and only if A = Â½ (B + I), where B is an idempotent matrix.

### Question 4: Does the inverse of an involutory matrix exist?

Solution:

Yes, an involutory matrix is invertible. The inverse of an involutory matrix is equal to the original matrix itself.

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