How to solve for $X$ in matrix equation $XA + X = 2B$?

Question

I am given homework where I have to solve for matrix $X$ equation $$B^{-1}XA = -B^{-1}X + 2E,$$ where $A$ and $B$ are known matrices and $E$ is the identity matrix.

I simplified it in the following way: $$B[B^{-1}XA] = B[-B^{-1}X + 2E]$$ $$\implies XA = -X + 2B$$ $$\implies XA + X = 2B$$

I might have done something wrong till now, if so please tell.

Though what I really want to know is: Does $X*A + X = X(A + 1)$ or $X*A + X = X(A + E)$ or something else?

Answer 1

On the one hand, you have $$X= X E = 1X,$$ because both multiplication by a scalar $1$ and by an identity matrix $E$ would result in the same thing.

However, the expression $A+1$ (where $A$ is a matrix) has no sense (except for the case where $A$ is a $1\times 1$ matrix, but even in this case we are pushing it), because $1$ is real number and we can't add it to a matrix - these objects are of different nature; remember, we don't add oranges to apples.

Therefore, for your equation, only the form $X(A+E)=2B$ is correct.

Answer 2

Yes, $XA + X = X(A+E) $ although $I $ is usually used to denote the identity matrix. The number 1 is a scalar, not a matrix.

Answer 3

The correct simplification is indeed $$ XA + X = 2B \implies\\ X(A + E) = 2B $$ as you can verify by distributing. Note that $A + 1$ is not a meaningful expression, since we can add a matrix to a scalar directly.