Open In App
Related Articles

Involutory Matrix

Improve
Improve
Improve
Like Article
Like
Save Article
Save
Report issue
Report

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 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 × 2.”

P_{2,2} = \left[\begin{array}{cc} 2 & 1\\ -3 & -2 \end{array}\right]

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

Q_{3,3} = \left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0 \end{array}\right]

involutory Matrix Formula

Let us consider a “2 × 2” square matrix A = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right]      . The given matrix is said to be an involutory matrix if satisfies the condition A2 = I

A^{2} = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right] \times \left[\begin{array}{cc} a & b\\ c & d \end{array}\right] = \left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right]

A^{2} = \left[\begin{array}{cc} a^{2}+bc & ab+bd\\ ac+cd & bc+d^{2} \end{array}\right] = \left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right]

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 A = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right]       is said to be an involutory matrix if 

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

Read More,

Solved Examples on involutory Matrix

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

A = \left[\begin{array}{ccc} 2 & 0 & 1\\ 0 & -1 & 0\\ -3 & 0 & -2 \end{array}\right]

Solution:

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

A^{2} = \left[\begin{array}{ccc} 2 & 0 & 1\\ 0 & -1 & 0\\ -3 & 0 & -2 \end{array}\right] \times\left[\begin{array}{ccc} 2 & 0 & 1\\ 0 & -1 & 0\\ -3 & 0 & -2 \end{array}\right]

A^{2} = \left[\begin{array}{ccc} (4+0-3) & (0+0+0) & (2+0-2)\\ (0+0+0) & (0+1+0) & (0+0+0)\\ (-6+0+6) & (0+0+0) & (-3+0+4) \end{array}\right]

A^{2} = \left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right] = I

Hence, verified.

So, the given matrix A is an involutory matrix.

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

Solution:

A matrix A = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right]       is said to be an involutory 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 involutory matrix is A = \left[\begin{array}{cc} 3 & 4\\ -2 & -3 \end{array}\right]      .

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

B = \left[\begin{array}{cc} 7 & 6\\ -8 & -7 \end{array}\right]

Solution:

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

B-1 = Adj B/ |B|

Adj B = \left[\begin{array}{cc} -7 & -6\\ 8 & 7 \end{array}\right]

|B| = −49 − (−48) = −1

B^{-1} = \frac{1}{-1}\left[\begin{array}{cc} -7 & -6\\ 8 & 7 \end{array}\right] = \left[\begin{array}{cc} -(-7) & -(-6)\\ -8 & -7 \end{array}\right]

B^{-1} = \left[\begin{array}{cc} 7 & 6\\ -8 & -7 \end{array}\right] = B

Hence, the given matrix is involutory.

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

Solution:

Let us consider of an involutory 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 involutory 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 involutory matrix of any order is always ±1.

Hence proved.

FAQs on involutory Matrix

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

Answer:

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 = ½ (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.



Last Updated : 08 Aug, 2023
Like Article
Save Article
Previous
Next
Share your thoughts in the comments
Similar Reads