# 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 $$a_1 \ne 0$$, 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).

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 $$x$$.