Is it possible to solve for a matrix using a system of equations?

Question

Let's say I have 3 linearly independent matrix-vector equations.

$A\mathbf{x_1} = \mathbf{b_1}$

$A\mathbf{x_2} = \mathbf{b_2}$

$A\mathbf{x_3} = \mathbf{b_3}$

Instinctively , I would try to set the equation like this

$\begin{bmatrix} A & 0 & 0 \\ 0 & A & 0 \\ 0 & 0 & A \end{bmatrix} \begin{bmatrix} \mathbf{x_1} \\ \mathbf{x_2} \\ \mathbf{x_3}\end{bmatrix} = \begin{bmatrix} \mathbf{b_1} \\ \mathbf{b_2} \\ \mathbf{b_3} \end{bmatrix}$

Is it possible to solve for the matrix $A$? And if so, how would I do that? Is there a name for this type of thing?

Answer 1

If $\{x_1,x_2,x_3\}$ is a base, then you have $$A\begin{pmatrix}x_1&x_2&x_3\end{pmatrix}=\begin{pmatrix}b_1&b_2&b_3\end{pmatrix}$$ or $$AX=B$$ then $$A=BX^{-1}$$

Answer 2

Yes there is. This process is called Gauss Elimination. Check it our here.

Using Gauss Elimination you can solve 3 equations of 3 variables.