Involutory matrix - Definition, Examples and its properties

SHARE:

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

Involutory matrix 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 is n x n identity matrix.

[ ##eye##  Idempotent matrix and its properties]

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 2 x 2 involutory matrix

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

Example of 3 x 3 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.( A) = Det. ( I )

So, Det.( A ) • Det.( A ) = 1

So,  Det.( A )= 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= 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 = 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 )= B•I•B ___[ A2 = for an Involutory matrix ]

AB )= B•B ______[ I•B ]

AB )= B2 = ___B2 = 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 = then the matrix A will be an Involutory matrix otherwise it won't be an Involutory matrix.

Read More Articles:

COMMENTS

BLOGGER: 3

Name

Electrical Basics,7,Electrical Machine,2,Electrical Q & A,2,Electronics,3,Instrumentation,2,Mathematics,2,Power Factor,3,Power System,7,
ltr
item
Electrical-Technology | All about Electrical Engineering: Involutory matrix - Definition, Examples and its properties
Involutory matrix - Definition, Examples and its properties
What is an Involutory matrix.?, Examples of 2X2 and 3X3 Involutory matrix. Properties of Involutory matrix. How to check Involutory matrix.
https://1.bp.blogspot.com/-tlRe6hKvtO4/YP2nu6ei2PI/AAAAAAAAAtM/5v1332PZ-0g1Skl1JR1CUl7C9IduRoqBACLcBGAsYHQ/w640-h272/What-is-Involutory-Matrix-and-its-Properties.webp
https://1.bp.blogspot.com/-tlRe6hKvtO4/YP2nu6ei2PI/AAAAAAAAAtM/5v1332PZ-0g1Skl1JR1CUl7C9IduRoqBACLcBGAsYHQ/s72-w640-c-h272/What-is-Involutory-Matrix-and-its-Properties.webp
Electrical-Technology | All about Electrical Engineering
https://www.electrical-technology.com/2021/07/definition-and-properties-of-Involutory-matrix.html
https://www.electrical-technology.com/
https://www.electrical-technology.com/
https://www.electrical-technology.com/2021/07/definition-and-properties-of-Involutory-matrix.html
true
1120548955437985836
UTF-8
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS PREMIUM CONTENT IS LOCKED STEP 1: Share to a social network STEP 2: Click the link on your social network Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy Table of Content