# What is an Orthogonal Matrix?-Example of Orthogonal Matrix

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

If AA^{t}=I

OR

A^{t}=A^{-1}

A = a square matrix

A^{t}=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:-**

Solution:

Given matrix

According to the definition of an orthogonal matrix

Prove that

We know that

Cos^{2} α+ sin^{2}α=1

-sinα cosα+sinα cosα=0

**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 A
^{t}is orthogonal - If A is an orthogonal matrix then A
^{n}is orthogonal - The product of two orthogonal matrices is also orthogonal

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