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 =
Here we observe the definition = = square root of (I
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 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 =
) = Det. ( I )
So, )• ) = 1
So, = 1
So, ) = square root ( 1 )
Hence, ) = ±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(+lA+AI +l )
= 1/4( I +lA+AI +l )__________ [ since l = l ]
= 1/4( 2•A + 2•l )_______ [ since lA=AI = A ]
= 1/2(A+I) = C
= 1/2(A+I).__________ [ Idempotent ] = C
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.
= • = l • I = I
= • =
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
= B•I•B ___[ =
= B•B ______[ I•B = B
= B = [ B =
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