Q: Prove that if $\displaystyle |A|=|B|\neq{0}$ and A and B are of the same size, then there exists a matrix C such that $\displaystyle |C|=1$ and A=CB.

A: Since $\displaystyle |A|=|B|\neq{0}$ we know $\displaystyle A^{-1}$ and $\displaystyle B^{-1}$ exists. So,

$\displaystyle |A^{-1}||A|=|A^{-1}||B|$

$\displaystyle 1=|A^{-1}||B|$

Now, let $\displaystyle |C|=1$

$\displaystyle |C|=|A^{-1}||B|$

Since $\displaystyle |A|=|B|$ and $\displaystyle |C|=|A^{-1}||A|$

$\displaystyle |A^{-1}||A|=|A^{-1}||B|$

$\displaystyle (|A||A^{-1}|)|A|=(|A||A^{-1}|)|B|$

$\displaystyle 1|A|=(|A||A^{-1}|)|B|$

$\displaystyle |A|=|C||B|$

$\displaystyle \therefore$ $\displaystyle A=CB$

Does that work? I guess A=I, B=I, and C=I. Do I need to show more?