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

Show that all ##n \times n## (real) orthogonal matrices ##O## leave invariant the quadratic form ##x_{1}^{2} + x_{2}^{2}+ \cdots + x_{n}^{2}##, that is, that if ##x'=Ox##, then ##x'^{2}=x^{2}##.

**2. Relevant equations**

**3. The attempt at a solution**

##x'^{2} = (x')^{T}(x') = (Ox)^{T}(Ox) = x^{T}O^{T}Ox = x^{T}x = x^{2}##.

I would like to check if I am correct?