# 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 involuntary 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 involuntary matrix that, when multiplied by itself, gives an identity matrix of the same order. A square matrix “P” is said to be an involuntary matrix if its inverse is the original matrix itself i.e. **P = P ^{-1}**.

### Examples of Involuntary Matrix

- The matrix given below is an involuntary matrix of order “2 Ã— 2.”

- The matrix given below is an involuntary matrix of order “3 Ã— 3.”

## Involuntary Matrix Formula

Let us consider a “2 Ã— 2” square matrix . The given matrix is said to be an involuntary matrix if satisfies the condition A

^{2}= INow, comparing the terms on each side, we get

a^{2}+ bc = 1ab + bd = 0

b (a + d) = 0

b = 0 or a + d = 0

d = âˆ’aSo, a square matrix is said to be an involuntary matrix if

- a
^{2}+ bc = 1- d = âˆ’a

## Properties of Involuntary Matrix

The following are some important properties of an involuntary matrix:

- A square matrix “A” of any order is said to be involuntary if and only if A
^{2}= I or A = A^{-1}. - If A and B are two involuntary matrices of the same order and AB = BA, then AB is also an involuntary matrix.
- The determinant of an involuntary matrix is always either -1 or +1.
- If “A” is an involuntary matrix of any order, then A
^{n}= I if n is even and A^{n}= A if n is odd, where n is an integer. - If a block diagonal matrix is derived from an involuntary matrix, then the obtained matrix is also involuntary.
- The eigenvalues of an involuntary matrix are always either -1 or +1.
- Symmetric involutory matrix is orthogonal, and vice versa.
- An involuntary matrix “A” can also be an idempotent matrix if “A” is an identity matrix.
- The following is the relationship between idempotent and involuntary matrices: A square matrix “A” is said to be an involuntary matrix if and only if A = Â½ (B + I), where B is an idempotent matrix.

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## Solved Examples on Involuntary Matrix

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

**Solution:**

To prove that the given matrix is involuntary, we have to prove that A

^{2}= A.Hence, verified.

So, the given matrix A is an involuntary matrix.

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

**Solution:**

A matrix is said to be an involuntary matrix, if a

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

^{2}+ bc = 1.(3)

^{2 }+ (4) Ã— (âˆ’2) = 9 âˆ’ 8 = 1We know that d = âˆ’a.

So, the involuntary matrix is .

**Example 3: Prove that the matrix given below is involuntary.**

**Solution:**

To prove that the given matrix is involuntary, we have to prove that B = B

^{-1}.B

^{-1}= Adj B/ |B||B| = âˆ’49 âˆ’ (âˆ’48) = âˆ’1

Hence, the given matrix is involuntary.

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

**Solution:**

Let us consider of an involuntary 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 involuntary if and only if P

^{2}= 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 involuntary matrix of any order is always Â±1.

Hence proved.

## FAQs on Involuntary Matrix

**Question 1: How to prove that a matrix is involuntary?**

**Answer:**

Any square matrix “P” is said to be an involuntary matrix if and only if P

^{2}= I or P = P^{-1}. So, to prove that a matrix is involuntary, the matrix must satisfy the above condition.

**Question 2: Define an involuntary matrix.**

**Solution:**

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

**Question 3: What is the relation between involuntary and idempotent matrices?**

**Solution:**

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

**Question 4: Does the inverse of an involuntary matrix exist?**

**Solution:**

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

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