How Can I Solve a Determinant Problem Using Induction?

[kidsmoker]

Homework Statement

Hi,
i'm trying to solve this problem:

http://img4.imageshack.us/img4/3876/53065718.jpg .[/URL]

The Attempt at a Solution

I have shown it for n=2 and n=3 then I was going to use induction to prove it for all n, but I can't seem to find a way to do it. Please help!

Thanks.

 

[CompuChip]

If you expand the determinant along the last column, then you will get terms of the form
$$a_i^n \begin{vmatrix} 1 & a_2 & a_2^2 & \cdots & a_2^{n-1} \\ 1 & a_3 & a_3^2 & \cdots & a_3^{n-1} \\ \vdots & \cdots & \cdots & \cdots & \cdots \\ 1 & a_{n} & a_n^2 & \cdots & a_n^{n-1} \\ \end{vmatrix}$$

 

[Architeuthis Dux]

Hello,

Thank you for reaching out for help with this determinant problem. It is great that you have already solved it for n=2 and n=3. Using induction to prove it for all n is a good approach, but it can be challenging to find the right way to do it. Here are some tips that may help you:

1. Start by writing out the general formula for the determinant of an n x n matrix. This will involve a sum of products of elements from the matrix.

2. Use the definition of a determinant to expand the formula. This means expanding along the first row, then the second row, and so on. This will give you a sum of products that you can manipulate algebraically.

3. Use the fact that the determinant of a matrix is equal to the determinant of its transpose. This means that you can swap rows and columns in the formula without changing its value.

4. Use the induction hypothesis to simplify the formula. This means assuming that the formula is true for n-1 and using it to simplify the original formula.

5. Finally, you should be left with a formula that is true for all n. You can prove this by showing that it satisfies the base case (n=2 or n=3) and that it satisfies the induction step (n-1).

I hope this helps you in solving the problem. If you have any further questions or need clarification, please don't hesitate to ask. Good luck!