Hello,
Unexpectedly, a Determinant is giving me trouble. We are given a Matrix in K^(n,n) where its diagonal stara from "1" and end in "n", and every other entry is a parameter a. (I will leave a picture in the bottom of my post.) My question is, how do I calculate its Determinant?
Here are a few things that I tried: I, obviously, tried to substract/add some coloms/rows to others, but to no avail. The best I could achieve is a row/colomn with n-2 zeros. After that I tried to use the Laplace-Theorem, but even the Sub Matrixes aren't looking nice at all and I can't find a result.
I also tried to take a^n outside, which works for every a=/=0, but after that I got instantly stuck and find myself in the same situation.
But somehow, you have to find a trivial instances, I feel.
After thinking more outside of the box, I even tried to guess the Determinant and proving it with induction, which may work, but its just so annoying to guess right and I feel like its cheating soing this way.
I hope someone can give me a lead.
Unexpectedly, a Determinant is giving me trouble. We are given a Matrix in K^(n,n) where its diagonal stara from "1" and end in "n", and every other entry is a parameter a. (I will leave a picture in the bottom of my post.) My question is, how do I calculate its Determinant?
Here are a few things that I tried: I, obviously, tried to substract/add some coloms/rows to others, but to no avail. The best I could achieve is a row/colomn with n-2 zeros. After that I tried to use the Laplace-Theorem, but even the Sub Matrixes aren't looking nice at all and I can't find a result.
I also tried to take a^n outside, which works for every a=/=0, but after that I got instantly stuck and find myself in the same situation.
But somehow, you have to find a trivial instances, I feel.
After thinking more outside of the box, I even tried to guess the Determinant and proving it with induction, which may work, but its just so annoying to guess right and I feel like its cheating soing this way.
I hope someone can give me a lead.
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