Involutory matrix – Definition, Examples and its properties

What is an Involutory matrix?

Definition: An Involutory matrix is simply a square matrix which when multiply itself will result in an identity matrix.

In other words, mathematically we can define involutory matrix as if A is a square matrix then matrix A will be called involutory matrix if and only if it satisfies the condition A² = I. Where I 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 3 x 3 Involutory matrix

Properties of Involutory matrix

As we have learned above that 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.( A² ) = Det. ( I )

So, Det.( A ) • Det.( A ) = 1

So,  Det.( A )² = 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)

       C² = 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 = I 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 )² = B•I•B ___[ A² = I for an Involutory matrix ]

( AB )² = B•B ______[ I•B = B ]

( AB )² = B² = I ___[ B² = I 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 Involutory 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 = I then the matrix A will be an Involutory matrix otherwise it won’t be an Involutory matrix.

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