Hello,

$\displaystyle a$ and $\displaystyle b$ are two column p-vectors

$\displaystyle A$ is a $\displaystyle p \times p$ square matrix.

Trying to proof something (see attachment) ... I had to force it a bit and even though it feels right, I do not know exactly how to apply commutative law on the following products so that they can cancel out with the division $\displaystyle \frac{ab^TA^{-1}}{b^TA^{-1}a}$ I might be wrong but I think they both produce the same $\displaystyle p \times p$ square matrix though looking at the original problem looks more like a scalar.

Basically I need to arrive to $\displaystyle \frac{ab^TA^{-1}}{ab^TA^{-1}}$ or $\displaystyle \frac{b^TA^{-1}a}{b^TA^{-1}a}$.

I was trying something along the lines of $\displaystyle (a)(b^TA^{-1})=(b^TA^{-1})^T(a)^T={A^{-1}}^Tba^T$ but I can't arrive to the desired result.

Maybe I am applying the commutative rules wrongly ... btw I don't remember the rules name for this otherwise I would just look it up ..

If they did cancel the output would be 1 or Identity_p?

TIA,

Best regards,

bravegag

This is the proof I am trying to make: