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Math Help - Commutative matrix

  1. #1
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    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.
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  2. #2
    Junior Member tedii's Avatar
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    Quote Originally Posted by noles2188 View Post
    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.
    Last edited by tedii; September 17th 2009 at 01:25 PM. Reason: typo
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