I'm given a matrix equation $XA = X - A$, where I know what $A$ is, how do I solve such a thing?

Question

Here's the matrix $A$:

\begin{pmatrix} -1 & 1 & 2\\ 2 & -2 & 3\\ -1 & 1 & -2 \end{pmatrix}

How do I solve the matrix equation $XA = X - A$?

The equation can be rewritten as $A = X(I-A)$. – player3236 Oct 24 '20 at 16:22 — player3236, Oct 24 '20 at 16:22

statnet22 · Answer 1 · 2020-10-24T23:28:43.787

2

Your equation is $XA = X - A$. This can be rewritten as follows:

$X - XA = A$

$X(I - A) = A$, where $I$ is the identity matrix.

Multiplying by the inverse of $I - A$ on the right gives:

$X(I-A)(I-A)^{-1} = A(I-A)^{-1}$,

giving you $X = A(I-A)^{-1}$.

You can then just compute the right hand side.

Hope this helps :)

edited Oct 24 '20 at 23:28

answered Oct 24 '20 at 16:23

statnet22

Alright, but what does I stand for? As: 1 0 0 0 1 0 0 0 1 ? – Ognjen Ognjanovic Oct 24 '20 at 16:33
Or an Idenity matrix rather – Ognjen Ognjanovic Oct 24 '20 at 16:33
$I$ is standard notation for the identity matrix. However this solution is incorrect, albeit only slightly. – player3236 Oct 24 '20 at 16:34
@player3236 I'll take a look at your Solution, thanks in general <3 – Ognjen Ognjanovic Oct 24 '20 at 16:47
It's $X-XA=A\to X(I-A)=A\to X=A(I-A)^{-1}$ finally we get $$X=\frac{1}{11} \left( \begin{array}{rrr} -5 & 5 & 9 \ 3 & -3 & 10 \ -1 & 1 & -7 \ \end{array} \right) $$ – Raffaele Oct 24 '20 at 17:03

1 Answers1