# Orthogonal matrices form a group

1. Nov 6, 2015

### spaghetti3451

1. The problem statement, all variables and given/known data

Show that the set of all $n \times n$ orthogonal matrices forms a group.

2. Relevant equations

3. The attempt at a solution

For two orthogonal matrices $O_{1}$ and $O_{2}$, $x'^{2} = x'^{T}x' = (O_{1}O_{2}x)^{T}(O_{1}O_{2}x) = x^{T}O_{2}^{T}O_{1}^{T}O_{1}O_{2}x = x^{T}O_{2}^{T}O_{2}x = x^{T}x = x^{2}.$

So, closure is obeyed.

Matrix multiplication is associative.

The identity element is the identity matrix.

$x'^{2} = (O^{-1}x)^{T}(O^{-1}x) = x^{T}(O^{-1})^{T}O^{-1}x = x^{T}(O^{T})^{-1}O^{-1}x = x^{T}(OO^{T})^{-1}x = x^{T}x = x^{2}$.

So, the inverse of any orthogonal matrix is an orthogonal matrix.