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 A^{2} = I. Where I is n x n identity matrix.
Here we observe the definition A^{2} = 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 viceversa.
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.

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 A^{2} = I.
Hence Det.( A^{2 }) = 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

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^{2 }= 1/2(A+I) • 1/2(A+I)
= 1/4(A+I) • (A+I)
= 1/4(A^{2}+lA+AI +l^{2})
= 1/4( I +lA+AI +l )__________ [ since l^{2} = l ]
= 1/4( 2•A + 2•l )_______ [ since lA=AI = A ]
= 1/2(A+I) = C
So C^{2} = C = 1/2(A+I).__________ [ Idempotent ]
Hence it proved that 1/2(A+I) is an idempotent matrix.

For an Involutory matrix A.
A^{n} = I___ if n is even natural number.
A^{n} = A___ if n is odd natural number.
Since A^{2} = I for an Involutory matrix
So A^{3} = I•A = A
A^{4} = A^{2} • A^{2} = l • I = I
A^{5} = A^{2} • A^{3} = I•A = A ___and so on.

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