find matrix element from matrix equation How to find "x" from this equation
$$
 \begin{bmatrix}
  a_1 & a_1^2 & \cdots & a_1^n \\
  a_2 & a_2^2 & \cdots & a_2^n \\
  \vdots  & \vdots& \ddots & \vdots \\
  a_m & a_m^2 & \cdots & a_m^n \\    
 \end{bmatrix}
 \begin{bmatrix}
  x \\
  b_2\\
  \vdots\\
  b_m \\    
 \end{bmatrix}
 =
 \begin{bmatrix}
  c_1\\
  c_2\\
  \vdots\\
  c_m \\    
 \end{bmatrix}
$$

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 $x$.
