Give two specific examples of a non-zero matrix X such that: AX = XA.

Question

If matrix A = \begin{bmatrix}3&1\\-2&-1\end{bmatrix}

I am trying to give two specific examples of a non-zero matrix X such that: AX = XA.

so far I let X = \begin{bmatrix}a&b\\c&d\end{bmatrix}

And so AX = XA, and I multiplied X and A respectivley to give the linear equations:

3a + c = 3a - 2b

3b + d = a - b

-2a + c = 3c - 2d

-2b - d = c - d

I believe i am supposed to show this in matrix form and solve the system to get a 4x4 matrix containing expressions with two variables where any number can be substituted so that AX = XA. However I am unsure on how to do this and need some help.

@TitoEliatron Certainly not. Multiplying on the left by a diagonal matrix has the effect of multiplying rows by the respective diagonal elements, while multiplying on the right has the same effect of columns. — Jean-Claude Arbaut, Nov 15 '18 at 21:06
Funny: similar question (and the same one day later), exactly one year ago: looks like a teacher is reusing the same exercise again and again, and students feel like asking on MSE again and again. — Jean-Claude Arbaut, Nov 15 '18 at 21:13
@Jean-ClaudeArbaut Actually, the user name has already been deleted, see above. — Dietrich Burde, Nov 15 '18 at 22:24

score 3 · Answer 1 · answered Nov 15 '18 at 21:04

3

The identity matrix always works, so that's one example.

$X=A$ always works, since if $X=A$, then $AX=A^2=XA$, so that's another one.

answered Nov 15 '18 at 21:04

quasi

58,772

score 0 · Answer 2 · answered Nov 15 '18 at 21:06

0

Any nonzero multiple, $t\cdot I$, of the identity matrix.

answered Nov 15 '18 at 21:06

Give two specific examples of a non-zero matrix X such that: AX = XA.

2 Answers2