[Math] Linear Algebra – Find Matrix X

inverselinear algebramatricesmatrix equations

Background Information:

I am studying linear algebra about matrix multiplication, addition, subtraction, and inverses. I came across this question that I have the book solution to, but my own solution is slightly different. I need your feedback about few steps that I am confused about, please go to "My Question" section.

Textbook Question:

Given the following matrices,

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find matrix X if any that satisfies the equation BX + AB = CX.

Textbook Solution:

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My Solution:

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My Question:

Notice the book solutions says the answer is (C – B)^-1 * AB, but my answer is AB * (C – B)^-1. I have calculated the result, and the answer is different. Could you please explain what am I doing wrong? Why does the book says (C – B)^-1 * AB is the answer?

Best Answer

We have that

$$BX+AB=CX \implies AB=CX-BX \implies(C-B)X=AB\\\implies (C-B)^{-1}(C-B)X=(C-B)^{-1}AB$$

and thus

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

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