Involutory Matrix

• Last Updated : 22 Feb, 2023

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 × n” matrix. For example, a matrix of order “5 × 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 × 2.”

• The matrix given below is an involuntary matrix of order “3 × 3.”

Involuntary Matrix Formula

Let us consider a “2 × 2” square matrix . The given matrix is said to be an involuntary 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 = −a

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

• a2 + bc = 1
• d = −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 A2 = 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 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 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 = ½ (B + I), where B is an idempotent matrix.

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 A2 = A.

Hence, verified.

So, the given matrix A is an involuntary matrix.

Example 2: Give an example of an involuntary matrix of order 2 × 2.

Solution:

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

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

(3)2 + (4) × (−2) = 9 − 8 = 1

We know that d = −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| = −49 − (−48) = −1

Hence, the given matrix is involuntary.

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

Solution:

Let us consider of an involuntary matrix “P” of order “n × n” to prove that its determinant is always ±1.

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

P × P = I

Now, |P| × |P| = |I|

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

(|P|)2 = 1

|P| = √1 = ±1

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

Hence proved.

FAQs on Involuntary Matrix

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

Any square matrix “P” is said to be an involuntary matrix if and only if P2 = 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 = ½ (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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