Involutory matrix – Definition, Examples and its properties

Involutory matrix and its Properties

What is an Involutory matrix?

An Involutory matrix is simply a square matrix that when multiplied itself will result in an identity matrix. In other words, mathematically we can define an involutory matrix as: If A is a square matrix then matrix A will be called an involutory matrix if and only if it satisfies the condition   A2 = I. Where is n x n identity matrix.

Here we observe the definition A2 = I, that is A = square root of (I).

It means the involutory matrix [A] is always the square root of an identity matrix [I].

Also, the size of an involutory matrix will be the same as the size of an identity matrix and vice-versa. Also, we can say that an Involuntary matrix is a square matrix that is its own inverse.

Examples of Involutory matrix

Example of 2 x 2 Involutory matrix

Example of 2 x 2 involutory matrix

Example of 3 x 3 Involutory matrix

Example of 3 x 3 involutory matrix

Properties of Involutory matrix

As we have learned above what is an involutory matrix, so let’s move forward and learn its important properties.

  1. The determinant of an Involuntary matrix will be either +1 or -1.

Let’s prove it with an example so that it will be easy to understand.

If A is a square matrix of size (n x n).

Then according to the definition of involutory matrix A2 = I.

Hence Det.( A) = Det. ( I )

So, Det.( A )• Det.( A ) = 1

So,  Det.( A )= 1

So, Det.( A )  = square root ( 1 )

Hence, Det.( A ) = ±1 = either +1 or -1

2. If A is ( n x n ) square matrix, then A will be involutory matrix if and only if 1/2(A+I) is an idempotent matrix.

Let C = 1/2(A+I)

      C= 1/2(A+I) • 1/2(A+I)

          = 1/4(A+I) • (A+I)

          = 1/4(A2+lA+AI +l2)

           = 1/4( I +lA+AI +l )__________ since l2 = l ]

           = 1/4( 2•A + 2•l )_______ since lA=AI = A ]

           1/2(A+I) = C

       So  C2 = C
              = 1/2(A+I).__________ [
Idempotent ]

Hence it proved that 1/2(A+I) is an idempotent matrix.

3. For an Involutory matrix A.
An = I___________ if n is even natural number.
An = A____________ if n is odd natural number.

Since A2 = for an Involutory matrix

So A3 = I•A = A

     A4 = A2 • A2 = l • I I

     A5 = A2 • A3 = I•A = A ___and so on.

4. If A and B are involutory matrices when AB = BA then AB will also, be an Involutory matrix.

Since AB = BA 

Multiply both sides by AB

So AB • AB = BA • AB

AB )= B•I•B ___[ A2 = for an Involutory matrix ]

AB )= B•B ______[ I•B ]

AB )= B2 = ___B2 = for an Involutory matrix ]

How to check whether a matrix is an Involutory matrix or not?

We can easily check whether any square matrix is Involuntary or not. For this find the square of that matrix and check the result whether you got the identity matrix or not.

If any square matrix A satisfies the condition A2 = then the matrix A will be an Involutory matrix otherwise it won’t be an Involutory matrix.

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Hey, I'm Satish Gupta an Engineer by profession and blogger by passion. I am writer and founder of this blog, Here I publish contents related to Electrical and Electronics Engineering..

1 thought on “Involutory matrix – Definition, Examples and its properties”

  1. Nice read, I just passed this onto a friend who was doing a little research on that. And he just bought me lunch as I found it for him smile Therefore let me rephrase that: Thanks for lunch!


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