Solving for unknown matrices. Specific example and useful rules.

Question

I wanted to ask how i could go about finding matrix X in the following equations:

AX-B^T= X + AB
and
AX=BX-2X+I

A and B are given in both examples. A,B and X are not the same in the 2 examples.

I realize that its supposedly quite basic,but i haven't been able to find a source online to pinpoint the rules of matrix arithmetic.

Thank you in advance! :)

Answer 1

Yes, it is easy. For example, rewrite the second equation as $(A-B+2I)X=I$. This means that $X=(A-B+2I)^{-1}$.

Answer 2

We have that

$$AX-B^T=X+AB\implies AX-X=AB+B^T\implies (A-I)X=AB+B^T.$$ Assuming $A-I$ is invertible (and I bet it is in your case) then

$$X=(A-I)^{-1}(AB+B^T).$$