Quote by superbro
If it's the zero matrix that is making you think of division by zero, it's also true for a 90 degree rotation in the opposite direction.
[itex]\left( \begin{array}{ccc} 0 & 1 \\ 1 & 0 \end{array} \right) x \cdot x = \left( \begin{array}{ccc} 0 & 1 \\ 1 & 0 \end{array} \right) x \cdot x[/itex]
Those matrices aren't equal, either.

The equation you've written is an example of the length of vectors being invariant under spatial rotations, which form a group. The rotation matrices are unimodular with orthogonal rows and columns.
To be clear, my confusion is that in several proofs the claim is made that two matrices are equal because of their relation via the dot product of any vector.

That is confusing.