find matrix element from matrix equation How to find "x" from this equation
$$
 \begin{bmatrix}
  a_1 & a_1^2 & \cdots & a_1^n \\
  a_2 & a_2^2 & \cdots & a_2^n \\
  \vdots  & \vdots& \ddots & \vdots \\
  a_m & a_m^2 & \cdots & a_m^n \\    
 \end{bmatrix}
 \begin{bmatrix}
  x \\
  b_2\\
  \vdots\\
  b_m \\    
 \end{bmatrix}
 =
 \begin{bmatrix}
  c_1\\
  c_2\\
  \vdots\\
  c_m \\    
 \end{bmatrix}
$$
 A: If $a_1 \ne 0$, just multiply the first row of the matrix by the vector $(x, b_2, \cdots, b_n)^T$ (you have a typo, it should be $b_n$, not $b_m$). 
$$
a_1 x + a_1^2 b_2 + \cdots + a_1^n b_n = c_1 \Leftrightarrow x = \frac{1}{a_1} \left(c_1 -\sum_{i=2}^n a_1^i b_i \right)
$$
Otherwise, just pick a row $k$ such that $a_k \ne 0$ and compute
$$
a_k x + a_k^2 b_2 + \cdots + a_k^n b_n = c_k \Leftrightarrow x = \frac{1}{a_k} \left(c_k -\sum_{i=2}^n a_k^i b_i \right)
$$
If all coefficients $a_k$ are zero, then either all coefficients $c_k$ are zero and any $x$ will do, or some coefficient $c_k$ is not zero and there is no solution.
[EDIT]
There is an issue that I did not address before but is also important. For every non zero coefficient $a_k$, you have an alternative expression for $x$. So, there will only be a solution if all these alternatives lead to the same value of $x$ (otherwise we will have incompatible equations).
A: If $n \ne m$, then the given equation makes no sense.
So, let $n=m$ and $j \in\{1,2,...,m\}.$ Then we have
$$a_jx+a_j^2b_2+...+a_j^mb_m=c_j.$$
If $a_j \ne 0$ we get
$$x= \frac{c_j}{a_j}-(a_jb_2+...+a_{j}^{m-1}b_m).$$
If $a_1=a_2=...=a_m=0$, then you can't find $x$.
