Finding and Proving Invertability of 2x2 Matrices

[daniel_i_l]

Homework Statement

a) find all the 2x2 matrices where AA=A.
b) prove that if
$$I + a_{1}A + a_{2}A^2 + ... + a_{k}A^k = 0$$
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:
$$A(a_{1}I + a_{2}A^1 + ... + a_{k}A^{k-1}) = -I$$
and if i take the determinant on each side i see that |A| <> 0 so it's invertable. Is that correct?
Thanks.

 

[radou]

b) looks correct.

Regarding a), assume A is regular and see what follows from that. Further on, consider A = I as a special case.

 

[Dick]

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.

 

[Dick]

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.