a.) If A2 = I, what are the possible eigenvalues of A?
b.) If this A is 2 by 2, and not I or -I, find its trace and determinant.
c.) If the first row is (3,-1), what is the second row?
None was given, but I think:
1. det(A) = 1 for A2 = I or A-1 = A
We're studying about Matrix diagonalization and the topic is called "Diagonalization of a Matrix". The equation S[tex]\Lambda[/tex]S-1 = A is supposed to be relevant.
The Attempt at a Solution
a.) I got the possible eigenvalues to be: [tex]\lambda[/tex]1 x [tex]\lambda[/tex] 2 x ... x [tex]\lambda[/tex]n = 1
b.) tr(A) = [tex]\lambda[/tex] + [tex]1/\lambda[/tex]
det(A) = 1
c.) This is where I'm stuck... I know I'm supposed to use the equation det(A) = 1 but there are two unknowns and I only know one equation. I was going to use the equation S[tex]\Lambda[/tex]S-1 = A but then I realized that I need the eigenvectors and I can't find the eigenvectors since I don't know the whole matrix.