1. ## Commutative matrix

Find all matrices that commute with the given matrix A. (The following is all one matrix).

[2 3]
[-3 2]

I let B = the matrix with four variables w, x (top row) y, z (bottom row). I then found AB and BA. I then had the following equations, and this is where I'm stuck.

2w + 3y = 2w - 3x
2x + 3z = 2x + 3w
2y - 3w = 2y - 3z
2z - 3x = 2z + 3y

Any knowledge on where to go from here would be great.

2. Originally Posted by noles2188
Find all matrices that commute with the given matrix A. (The following is all one matrix).

[2 3]
[-3 2]

I let B = the matrix with four variables w, x (top row) y, z (bottom row). I then found AB and BA. I then had the following equations, and this is where I'm stuck.

2w + 3y = 2w - 3x
2x + 3z = 2x + 3w
2y - 3w = 2y - 3z
2z - 3x = 2z + 3y

Any knowledge on where to go from here would be great.
Simplify all equations and set the variables equal to zero.
You can eliminate the third and fourth row because it's the 'same' as the first and second when simplified.

You get:

y+x=0
z-w=0

=>

Any matrix [w, x; y, z] (where ; is the next row) can commute with matrix A if y+x=0 and z-w=0 or y=-x and z=w.