You don't have a problem. Note that your eigenvectors are just multiples of the book's answer. This is fine!
(i) Determine all the eigenvalues of A
(ii) For each eigenvalue of A, find the set of eigenvectors corresponding to
A = ( 1 2 )
( 3 2 )
I found eigenvalues of A to be 4 and -1.
I also found the eigenvectors to be (2/3,1) for =4 and (-1,1) for =-1
BUT the solution in the back of the book says (2,3) for =4 and (1,-1) for =1
I'm soo confusedd! Can someone tell me what's wrongg? Am I calculating the eigenvectors wrong?
Here's how i calculate eigenvector for =4
A-4I = (-3 2)
(3 -2)
then (-3 2 |0)
(3 -2 |0)
and i end up with x1 -(2/3) x2 =0. so x2 = t, and x1 = (2/3)t. eigenvector = (2/3,1) but book says (2,3)
and for =-1
A-(-1)I = (2 3 )
(2 3 )
then (2 3 |0)
(2 3 |0)
and i end up with x1+x2=0. x2= t and x1 = -t. eigenvector = (-1,1) but book says (1,-1)
The set of all eigenvectors of a matrix A, corresponding to a given eigenvalue, , form a subspace. In particular, any multiple of an eigenvector is also an eigenvector, corresponding to the same eigenvalue: If and "r" is any scalar, then .