If A,B,C,D are matrices such that C and D commute and D is invertible, then
There is a hint that says " multiply on the right by ".
So I observed that:
If we choose , then
Now since C and D commute, . Thus
It looked like a convincing proof until I realised that I have used the hypothesis in the penultimate step. I have also used it while setting .
So how can I fix those steps? Or do we need a different direction?
Assume are all square and multiplicativly conformant (I believe that is implied by the conditions in the problem)
expanding the first determinant in the last expression, from the bottom right gives: