Why can I cancel out the matrix in the middle

Question

I was given an example from my teacher when he does something like

$$A^TC^{-1}A \gt B^TC^{-1}B \implies A^TA \gt B^TB$$

Where A and B are vectors.

I am curious on how can the inversed matrix C in the middle got cancelled out?

Answer 1

If that's really what your teacher did, it's a mistake.

Try e.g. $$ A = \pmatrix{1\cr 0\cr},\ B = \pmatrix{0\cr 2\cr},\ C^{-1} = \pmatrix{5 & 0\cr 0 & 1\cr}$$