I'm wondering if a specific matrix is invertible or not. More specifically, let $a_{1} = \begin{pmatrix} 1\\ 2\\ 3\\ 4\\ 5 \end{pmatrix}$. Consider a $5\times5$ matrix $A$ such that $A=[a_1, (a_1)^2, (a_1)^3, (a_1)^4, (a_1)^5]$, where this powers are element-wise powers.
So $A=\begin{pmatrix} 1 & 1& 1& 1 &1 \\ 2 & 4 & 8 & 16 &32 \\ 3 & 9 & 27 & 81 &243 \\ 4 & 16& 64& 256 & 1024\\ 5 & 25& 125& 625 & 3125 \end{pmatrix}$.
In this case, I know $A$ is invertible. For an arbitrary $n\times n$ matrix A, where $A=\begin{pmatrix} a_{1} &\cdots &(a_{1})^n \end{pmatrix}$, how can we tell A is invertible or not? All elements in $a_{1}$ are distinct.
Your matrix will be invertible whenever the elements $a_i$ are all distinct and non-zero. This follows from the invertibility of the Vandermonde matrix. In particular, it suffices to note that your matrix can be written as the product $M = DV$ where $D$ is the diagonal matrix $$ D = \pmatrix{a_1\\&a_2\\ &&\ddots\\ &&& a_n} $$ and $V$ is the Vandermonde matrix, as defined in the linked article.
Of course, if one of the $a_i$ is equal to zero, then the matrix will have a row of zeros and therefore fail to be invertible.