# Finding two different matrices that have the same product with another one? (AB=AC)

Say you have a matrix, $A$, and you want to find two matrices $B, C$ for which $AB=AC$, but being $B\neq C$. I know this is quite possible due to matrices' nature, but I can't quite find a method to find them except for just random search.

i.e. Say $A = \begin{pmatrix} 1&1&1&1\\ 1&2&3&4\\ 2&4&4&6 \end{pmatrix}$

You can probably use some linear combination trick, either with rows or columns; or by creating a template matrix with a parameter that doesn't intervene on the final result, and hence you would be able to find infinite examples; but I'm unable to find a method for either of these ways.

I'm not looking for a particular solution, but rather for a method that would allow you to find them, I don't mind the result as long as I learn how to do it.

Anyway, thanks in advance. Any help appreciated.

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You need to find a matrix whose column vectors are in the nullspace of $A$. Say that $\{v_1,\dots,v_n\}$ span the nullspace of $A$. Then, choose any matrix $D$ and construct any matrix $E$ whose column vectors are linear combinations of $\{v_1,\dots,v_n\}$ (you need to pick $D$ and $E$ to be of appropriate dimension). Then,
$$A(D+E)=AD+AE=AD.$$
This gives you your "infinite source" of $B$s and $C$s. You can compute $v_1,\dots,v_n$ simply solving a system of linear equations, see here.