What is Orthogonal Matrix? Determinant and Examples

Orthogonal matrix is a square matrix R=[xij] such that RT = R-1. In other words, a square matrix (R) whose transpose is equal to its inverse is known as orthogonal matrix i.e. RT = R-1.

Orthogonal matrix examples

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

    \[I=\begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}\]

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

    \[I=\begin{bmatrix} 1 &0 &0 \\ 0& 1 &0 \\ 0& 0 & 1 \end{bmatrix}\]

Orthogonal matrix properties

1. Let B be an orthogonal matrix then BBT = BTB = 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 BBT = I (according to the property)

BBT = I

⇒ |BBT| = |I| = 1

⇒ |B| |BT| = 1

⇒ (|B|)2 = 1

⇒ |B| = ±1

Hence proved.

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