Thread: commutative law of the product of two p vectors and pxp matrix

1. commutative law of the product of two p vectors and pxp matrix

Hello,

$a$ and $b$ are two column p-vectors
$A$ is a $p \times p$ 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 $\frac{ab^TA^{-1}}{b^TA^{-1}a}$ I might be wrong but I think they both produce the same $p \times p$ square matrix though looking at the original problem looks more like a scalar.

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

I was trying something along the lines of $(a)(b^TA^{-1})=(b^TA^{-1})^T(a)^T={A^{-1}}^Tba^T$ 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:

2. Already found the solution ... only needed to realize that e.g. $b^TA^{-1}a$ is a scalar then I can move it around and solve the proof.