What is an Involutory matrix.?, Examples of 2X2 and 3X3 Involutory matrix. Properties of Involutory matrix. How to check Involutory matrix.

**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: I*f 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.**[ ##eye## Idempotent matrix and its properties]**

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**

**[ ##eye## Theory and working of Star-Delta Starter]**

**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 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

**]**

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

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Very good information

ReplyDeleteNice article

ReplyDeleteWell explained

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