Idempotent Matrices: Proving PQ+QP=0 implies PQ=QP=0

[td21]

Homework Statement

P,Q is idempotent, i.e. P^2=P and Q^2=Q.
Show that: If PQ+QP=0, PQ=QP=0.

Homework Equations

The Attempt at a Solution

PQ+QP=0
PQ=-QP
PPQ=-PQP
PPQP=-PQPP
PQP=-PQP
PQP=0

I can only show PQP=0, but how to show PQ=QP=0?
Thanks.

 

[Dick]

Homework Statement

P,Q is idempotent, i.e. P^2=P and Q^2=Q.
Show that: If PQ+QP=0, PQ=QP=0.

Homework Equations

The Attempt at a Solution

PQ+QP=0
PQ=-QP
PPQ=-PQP
PPQP=-PQPP
PQP=-PQP
PQP=0

I can only show PQP=0, but how to show PQ=QP=0?
Thanks.

I really don't think you want me to tell you. You are SO close. Look at the third line where you showed PPQ=-PQP.

 

[td21]

I really don't think you want me to tell you. You are SO close. Look at the third line where you showed PPQ=-PQP.

Thanks! But i still have trouble...
From 3rd line:
PPQ=-PQP
then
PQ=-PQP
PQQ=-PQPQ
PQ=-PQPQ
PQ(PQ+I)=0
PQ=0 or PQ=-I

Similarly,QP=0 or PQ=-I

Conclusion:
PQ=QP=0 or PQ=QP=-I

but still cannot show PQ=QP=0 ??

 

[Dick]

Thanks! But i still have trouble...
From 3rd line:
PPQ=-PQP
then
PQ=-PQP
PQQ=-PQPQ
PQ=-PQPQ
PQ(PQ+I)=0
PQ=0 or PQ=-I

Similarly,QP=0 or PQ=-I

Conclusion:
PQ=QP=0 or PQ=QP=-I

but still cannot show PQ=QP=0 ??

You completely overshot the goal. You showed PQ=(-PQP) and PQP=0. Just substitute the second into the first. So PQ=(-0)=0.