**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: I*f A is a square matrix then matrix A will be called an 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 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 what is an involutory matrix, so let’s move forward and learn its important properties.

The determinant ofan 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**

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 ^{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

= 1/2(A+I).__________ [

**]**

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

3. 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.

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**___[

**A**=

^{2}**I**for an Involutory matrix ]

**( ****AB ) ^{2 }**

**= B•B**______[

**I•B**=

**B**]

**( ****AB ) ^{2 }**

**=**

**B**=

^{2}**I**___[

**B**

^{2}=

**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 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 **A**^{2} = **I **then the matrix **A** will be an **Involutory matrix** otherwise it won’t be an Involutory matrix.

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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!