Reformulate your set of equations such that the coefficients are the unknowns and perform least square fitting on A.

This can be done by the following trick:

Be $x$ a column vector $(n\times 1)$, $A$ the unknown matrix $(n\times n)$, you know that $Ax =b \Rightarrow \sum_j a_{ij} x_j = b_i$ but this can also be seen as $\hat X\, \textrm{Vec}(A)= b$ where $\textrm{Vec}(A)$ is the vectorization operator and $\hat X$ must be n-times repeated and shifted by n $x^T$ or $0^T$ otherwise, leading to $\Rightarrow \hat X=x^T\otimes I_n $, where $I_n$ is the identity matrix of dimension n.

For the full system you can rewrite it as following, set $X=(x_1,...,x_n)$ and $B=(b_1,...,b_n)$ of your original equations $A X = B$, your least square problem for $A$ reads $(X^T\otimes I_n)\textrm{Vec}(A)=\textrm{Vec}(B^T)$. (modulo some transpose, I might have missed.)

So it all boils down if you have enough matrix equations whether this will be an over or undertemined system and can be directly solved by standard numerical techniques.

Reformulate your set of equations such that the coefficients are the unknowns and perform least square fitting on A.

This can be done by the following trick:

Be $x$ a column vector $(n\times 1)$, $A$ the unknown matrix $(n\times n)$, you know that $Ax =b \Rightarrow \sum_j a_{ij} x_j = b_i$ but this can also be seen as $\hat X\, \textrm{Vec}(A)= b$ where $\textrm{Vec}(A)$ is the vectorization operator and $\hat X$ must be n-times repeated and shifted by n $x^T$ or $0^T$ otherwise, leading to $\Rightarrow \hat X=x^T\otimes I_n $, where $I_n$ is the identity matrix of dimension n.

For the full system you can rewrite it as following, set $X=(x_1,...,x_n)$ and $B=(b_1,...,b_n)$ of your original equations $A X = B$, your least square problem for $A$ reads $(X^T\otimes I_n)\textrm{Vec}(A)=\textrm{Vec}(B)$.

So it all boils down if you have enough matrix equations whether this will be an over or undertemined system and can be directly solved by standard numerical techniques.

Reformulate your set of equations such that the coefficients are the unknowns and perform least square fitting on A.

This can be done by the following trick: be x a column vector $(n\times 1)$, A the matrix $(n\times n)$, you know that $Ax =b \Rightarrow \sum_j a_{ij} x_j = b_i$ but this can also be seen as $\hat X\, \textrm{Vec}(A)= b$ where $\textrm{Vec}(A)$ is the vectorization operator and $\hat X$ must be n-times repeated $x^T$ shifted by n or zero otherwise, leading to $\Rightarrow \hat X=x^T\otimes I_n $, where $I_n$ is the identity matrix of dimension n.

So it all boils down if you have enough matrix equations whether this will be an over or undertemined system and can be directly solved by standard numerical techniques.

Reformulate your set of equations such that the coefficients are the unknowns and perform least square fitting on A.

This can be done by the following trick:

Be $x$ a column vector $(n\times 1)$, $A$ the unknown matrix $(n\times n)$, you know that $Ax =b \Rightarrow \sum_j a_{ij} x_j = b_i$ but this can also be seen as $\hat X\, \textrm{Vec}(A)= b$ where $\textrm{Vec}(A)$ is the vectorization operator and $\hat X$ must be n-times repeated and shifted by n $x^T$ or $0^T$ otherwise, leading to $\Rightarrow \hat X=x^T\otimes I_n $, where $I_n$ is the identity matrix of dimension n.

For the full system you can rewrite it as following, set $X=(x_1,...,x_n)$ and $B=(b_1,...,b_n)$ of your original equations $A X = B$, your least square problem for $A$ reads $(X^T\otimes I_n)\textrm{Vec}(A)=\textrm{Vec}(B^T)$. (modulo some transpose, I might have missed.)

So it all boils down if you have enough matrix equations whether this will be an over or undertemined system and can be directly solved by standard numerical techniques.