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For this $n \times n$ matrix I am trying to find the determinant:

$\left[\begin{array}{cccc} 1 & 2 & ... & n \\ 1 & 2^3 & ... & n^3 \\ . & . & ... & . \\ 1 & 2^{2n-1} & ...& n^{2n-1} \end{array}\right]$

I am having abit of trouble with row reduction hence not getting anywhere. I think I need to transform it somehow as it looks similar to a vandermonde matrix but I am not sure how can someone help me out?

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  • $\begingroup$ What's the determinant when $n = 1, 2, 3, 4, 5$? $\endgroup$ – Calvin Lin Dec 6 '20 at 10:06
  • $\begingroup$ @CalvinLin 1,6,720,3628800, then they get really big... $\endgroup$ – user858070 Dec 6 '20 at 10:16
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(Fill in the gaps as needed. If you're stuck, show your work and explain why you are stuck.)

Hint: Show that the matrix whose entries $a_{ij} = b_i ^j$ has determinant $ \prod_{i > j } ( b_i - b_j)$.

Hint: Hence, conclude that the answer is $n! \prod_{i > j} ( i^2 - j^2 ) $.
This matches up with your calculated values when $ n = 1, 2, 3, 4$.

Note: The way the numerical answer is written up is the hint of what matrix you should be considering.

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