perplexus.info :: Just Math : Subtract Products, Get Identity?

Subtract Products, Get Identity? (Posted on 2007-04-11)

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

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

Submitted by K Sengupta

Rating: 2.5000 (2 votes)

Solution: (Hide)

             p₁₁ p₁₂
Let, P =
             p₂₁ p₂₂

               q₁₁ q₁₂
And, Q =
                q₂₁ q₂₂

               r₁₁ r₁₂
And, R =
                r₂₁ r₂₂

where:

Q = PQ - QP

Then r_11
= p_11 q_11 + p_12 q_21 - q_11 p_11 - q_12 p_21
= p_12 q_21 - q_12 p_21

and r_22
= p_21 q_12 + p_22 q_22 - q_21 p_12 - q_22 p_22
= p_21 q_12 - q_21 p_12

or r_11 = - r_22.

Accordingly, r_11 and r_22 cannot both be equal to 1. This is a contradiction.

Thus, it is NEVER the case that PQ - QP = I

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
general case	Stefan	2007-04-26 04:45:01
re: Solution and a variant	Gamer	2007-04-11 19:25:43
Solution and a variant	Jer	2007-04-11 13:36:30

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