Linear algebra

  • Thread starter daniel_i_l
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  • #1
daniel_i_l
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Homework Statement


a) find all the 2x2 matrices where AA=A.
b) prove that if
[tex]
I + a_{1}A + a_{2}A^2 + ... + a_{k}A^k = 0
[/tex]
then A is invertable


Homework Equations


1)det(A) = 0 iff A isn't invertable


The Attempt at a Solution


a) I'm not sure how to approch this. I found that if A is invertable then the only solution is A=I but how do i cover the other cases?

b) by rearanging:
[tex]
A(a_{1}I + a_{2}A^1 + ... + a_{k}A^{k-1}) = -I
[/tex]
and if i take the determinant on each side i see that |A| <> 0 so it's invertable. Is that correct?
Thanks.
 
Last edited:

Answers and Replies

  • #2
radou
Homework Helper
3,115
6
b) looks correct.

Regarding a), assume A is regular and see what follows from that. Further on, consider A = I as a special case.
 
  • #3
Dick
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619
AA=A says that A is a projection operator. So consider what the range of A is. The 'other' case you are after is where it is a one-dimensional subspace of R^2. BTW, you don't have to show det(A) is non-zero, you've constructed an explicit inverse.
 
  • #4
Dick
Science Advisor
Homework Helper
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You could also just set A=[[a,b],[c,d]] and write the condition A^2=A. You can then eliminate two of the variables (except for singular cases). There are a two parameter family of these babies.
 

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