How to isolate matrix of interest from equation?

Question

I have an equation in the form of $A = BX - CX$, where $A, B, C$ are matrices. I would like to isolate $X$.

So far, I can't think of any elementary ways by following the basic rules of matrix operations, so I would appreciate any leads!

Answer 1

Matrices follow a distributive law from either the left, or the right. In some cases, you can distribute from either side and still have an equivalent matrix, but those are special cases and may not apply here.

For your question, if:

$$A=BX-CX$$

Then we can factor out $X$:

$$A=(B-C)X$$

which is the inverse of right distribution.

Now, in order for the first equation to even make sense, $X$ has to be multiplication compatible with both $B$ and $C$. Therefore, $B$ and $C$ have to compatible through addition. So, assuming that $(B-C)$ is a square and invertible/nonsingluar matrix, then we can multiply on the left by the inverse:

$$A=(B-C)X$$ becomes $$(B-C)^{-1}A=X$$

Answer 2

$A=(B-C)X\implies X=(B-C)^{-1} A$ provided $(B-C)$ is invertible.