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Linear algebra proof.

  • #1

Homework Statement


If A is an invertible idempotent matrix, then A must be the Identity matrix I_n.


Homework Equations


A^2==A ; A^2==AA; A^(-1); I==A^(-1)


The Attempt at a Solution



Suppose A is an nxn matrix =/= I_n.

s.t. A^(2)==A

so A^(2)==A ==> AA==A

==> A^(-1)AA==A^(-1)A ==> A==I==> A^(-1)A==A^(-1)I==>I==A^(-1)I==A^(-1)==A

which yeilds a contradiction because we supposed our A =/= I_n.

Therefore A==I_n


Is this correct please help me understand where I have failed...
 

Answers and Replies

  • #2
vela
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Suppose A is an nxn matrix =/= I_n.

s.t. A^(2)==A

so A^(2)==A ==> AA==A
==> A^(-1)AA==A^(-1)A
==> A==I
You should have stopped right here. You should have also stated that A is invertible.
==> A^(-1)A==A^(-1)I
==> I==A^(-1)I==A^(-1)==A
This is wrong. You don't know that the inverse of A is equal to A.
 
  • #3
Okay so if i stated A an invertible nxn matrix =/= I_n

s.t A^(2)==A(idempotent)... truncating the last bit of foolishness. I was correct?
 
  • #4
another proof...
problem statement.
prove if A and B are idempotent and AB==BA then AB is idempotent.

AB==BA ==> A^(-1), B^(-1) exist

Since A and B are idempotent invertible matrices, from previously proven theorem, we know A=I and B=I. and since II==I ==> AB==I Therefore AB==BA and AB is Idempotent,
 
  • #5
vela
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Education Advisor
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AB==BA ==> A^(-1), B^(-1) exist
This isn't true.
 

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