Hi everybody! I need some help with this matrix equation:

$$\pmatrix{6 & 1 \cr -3 & -8} -X \pmatrix{1 & 6 \cr -1 & 5}=I$$

So this is how I would solve it:

$$A-X\cdot B=I

\Leftrightarrow -X=I\cdot B^{-1}-A$$

Is this correct?

No, it's not. In fact, it would not be correct even if these were numbers. From a- xb= 1, multiplying by \(\displaystyle b^{-1}\) would give \(\displaystyle ab^{-1}- x=b^{-1}\) so that \(\displaystyle x= ab^{-1}- b^{-1}\). With A, B, X matrices, it is exactly the same with \(\displaystyle X= AB^{-1}- B^{-1}= (A- I)B^{-1}\).

Also, is -X different from X when it comes to matrices? Should I just see it as -1 times the matrix X, so then I would multiply all the numbers in the matrix with -1? I appreciate any help I can get with this!

-X is different from X in

**any** algebraic system as long as X is not 0! Yes, you can interpret -X as -1 times X which would be all entries of X multiplied by -1. It is also the same as "X subtracted from the 0 matrix" as well as the diagonal matrix with all diagonal entries "-1" times X.