**Orthogonal matrix** is a square matrix R=[x_{ij}] such that R^{T} = R^{-1}. In other words, a square matrix (R) whose transpose is equal to its inverse is known as orthogonal matrix i.e. R^{T} = R^{-1}.

**Contents**show

## Orthogonal matrix examples

The best example of an orthogonal matrix is an identity matrix or unit matrix as shown below.

The example of an orthogonal matrix of order 3 (or matrix size is 3 x 3) is given as

## Orthogonal matrix properties

1. Let B be an orthogonal matrix then BB^{T} = B^{T}B = I, where I is the identity matrix.

## Orthogonal matrix determinant

The determinant of an orthogonal matrix is always has a modulus of 1.

**Proof:**

Let B be an orthogonal matrix then BB^{T} = I (according to the property)

BB^{T} = I

⇒ |BB^{T}| = |I| = 1

⇒ |B| |B^{T}| = 1

⇒ (|B|)^{2} = 1

⇒ |B| = ±1

Hence proved.