How to fin these determinants?

[monkeydog]

How to find these determinants?

\(\displaystyle \begin{vmatrix}
1 & 1 & 1 & \cdots & 1 \\
a_{1} & a_{2} & a_{3} & \cdots & a_{n} \\
a_{1}^{2} & a_{2}^{2} & a_{3}^{2} & \cdots & a_{n}^{2} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a_{1}^{n-1} & a_{2}^{n-1} & a_{3}^{n-1} & \cdots & a_{n}^{n-1}
\end{vmatrix}
\)

\(\displaystyle \begin{vmatrix}
1 & 2 & 2 & \cdots & 2 \\
2 & 2 &2 & \cdots & 2 \\
2 & 2 & 3 & \cdots & 2 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
2 & 2 & 2 & \cdots & n
\end{vmatrix}
\)

Thank you very much!

 

[tkhunny]

What happens to the value of the determinant if you replace column 2 by the difference (column 2 less column 1)?

 

[daon2]

The first is called the Vandermonde matrix. There are several ways to obtain it, but following tkhunny's first step is the most direct. Of course if any of the a_i's are the same, the answer is 0. The answer, which is obtainable on Wikipedia, is the product of all \(\displaystyle a_i-a_j\) where i>j.

You will also need to use the facts that (1) if E is the elementary matrix dvividing a row of A by a nonzero constant k, then \(\displaystyle k\det EA = \det A\) and (2) adding a multiple of one row to another does not change the determinant. Co-factor expansion will also be used.