P and Q are two 2x2 matrices and I is a 2x2 unit matrix.

Is it ever the case that PQ - QP = I?

Matricies formatted as best I could:

    |a b|
P = |c d|

Q = |e f|
    |g h|

PQ - QP = |(bg-cf)       (af+bh-be-df)|
          |(ce+dg-ag-ch)       (cf-bg)|

To equal I, (bg-cf) = 1 = (cf-bg) which is clearly impossible.

So it is never the case that PQ - QP = I

An error in my first attempt led to

P = |-2 1|
    | 2 2|

Q = |1  1|
    |1 -1|

And PQ + QP = -I

Posted by Jer on 2007-04-11 13:36:30