About finding the inverse of a matrix I am solving a linear algebra problem and this matrix came up from a system of linear equations.

$A = \begin{pmatrix}
1 &amp; 2 &amp; \cdots &amp; n \\
1 &amp; 2^2 &amp; \cdots &amp; n^2 \\
\vdots &amp;  &amp; &amp;\vdots &amp;\\
1 &amp; 2^n &amp; \cdots &amp; n^n
\end{pmatrix}
$

I do not know how to check if my system has an unique solution or not (or the matrix is invertible) in this case.
The way to go is to run, say, LU-factorisation using one of broadly available codes allowing evaluation of the matrix condition ratio. Run the program for n=2,3,4... to see the pattern. You will probably see that the condition ratio is growing with each next n (in numerical sense, the matrix looks "bad" to me in general). As soon as the condition ratio reaches the value of about 10**16, the matrix (and all the others for n larger than the found one) is singular in numerical sense, because the computer operates with double precision variables, ie. it stores 16 significant digits only.
The way to go is to run, say, LU-factorisation using one of broadly available codes allowing evaluation of the matrix condition ratio. Run the program for n=2,3,4... to see the pattern. You will probably see that the condition ratio is growing with each next n (in numerical sense, the matrix looks "bad" to me in general). As soon as the condition ratio reaches the value of about 10**16, the matrix (and all the others for n larger than the found one) is singular in numerical sense, because the computer operates with double precision variables, ie. it stores 16 significant digits only.