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What is an Orthogonal Matrix?-Example of Orthogonal Matrix

July 28, 2022
written by Azhar Ejaz

A square matrix ‘A’ over R is called an orthogonal matrix

If AAt=I

OR

At=A-1

A = a square matrix

At=transpose of A

I=identity matrix of the same order as ‘A’

A-1=inverse of A

The product of the square matrix and it is transposed gives an identity matrix same order is called an orthogonal matrix. Or we can say A square matrix with real numbers is called an orthogonal matrix if its transpose is equal to its inverse of a matrix.

For example:-

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Solution:

Given matrix

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According to the definition of an orthogonal matrix

Prove that

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 We know that

Cos2 α+ sin2α=1

-sinα cosα+sinα cosα=0

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Summary

  • Every identity matrix is an orthogonal matrix
  • If A is an orthogonal matrix then A-1 is also an orthogonal matrix
  • If A is an orthogonal matrix then At is orthogonal
  • If A is an orthogonal matrix then An is orthogonal
  • The product of two orthogonal matrices is also orthogonal