Calculating an n X n determinant

TGV320
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Homework Statement
Help in order to solve a determinant
Relevant Equations
Determinants
Hello,

I need some advice because I just can't figure out how to solve the problem. I could try to make the determinant triangular by adding all the b together, but that doen't seem a good way of solving the problem. Is there any direction I should be thinking of?

1699948294438.jpg
Thanks
 
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Why not calculate the determinant for ##n = 2, 3, 4## and see whether a pattern emerges?
 
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TGV320 said:
Homework Statement: Help in order to solve a determinant
Relevant Equations: Determinants

Hello,

I need some advice because I just can't figure out how to solve the problem. I could try to make the determinant triangular by adding all the b together, but that doen't seem a good way of solving the problem. Is there any direction I should be thinking of?

View attachment 335323Thanks
Hint: Follow PeroK's advice and find the determinant by expanding along the bottom row.

-Dan
 
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By considering the Leibniz formula, one can figure out that only some terms survive, where the permutations do not contain zeroes.
 
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Multiply the ##i^{th}## row by ##-a_i## and add it to the first. You just need to see what the top left element will be.
 
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Hi,
Thanks for the advice.
I have figured it out,though I never thought about getting the answer by experimenting on it, always thought it to n. That way of doing it with n=2 then 3 is quite illuminating.

1700045361111.jpg
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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