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Subtract Products, Get Identity? (Posted on 2007-04-11) Difficulty: 3 of 5
P and Q are two 2x2 matrices and I is a 2x2 unit matrix.

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

See The Solution Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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Solution Solution and a variant | Comment 1 of 3
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
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