Question

Use Exercise 31 to find

$\mathit{d}\mathit{e}\mathit{t}\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 2& 3& 4& 5\\ 1& 4& 9& 16& 25\\ 1& 8& 27& 64& 125\\ 1& 16& 81& 256& 625\end{array}\right]$

Do not use technology.

Short answer

Therefore, the determinant of the given matrix is given by,

$detA=288$.

Step-by-step solution

By using Vandermonde’s Formula. 

For $n=1$, we have a $2\times 2$matrix

$A=\left[\begin{array}{cc}1& 1\\ a_0& a_1\end{array}\right]$

Using Vandermonde's formula, we have

$\underset{}{\prod _{i>j}\left(a_i-a_j\right)=a_1-a_0=detA}$.

To find the determinant.

Let $a_0=1,a_1=2,a_2=3,a_3=4,a_4=5$.

Now we can use Vandermonde's formula to find this determinant.

We compute,

$\begin{array}{l}detA=\prod _{\frac{i,j-1}{i>j}}(a_i-a_j)\\ detA=\left(2-1\right)\left(3-1\right)\left(4-1\right)\left(5-1\right)\left(3-2\right)\left(4-2\right)\left(5-2\right)\left(4-3\right)\left(5-3\right)\left(5-4\right)\\ detA=288\end{array}$