# Homework Help: Linear Algebra Homework Question

1. Dec 12, 2011

### simplyderp

1. The problem statement, all variables and given/known data

Prove that A=[a,b;c,d]
is diagonalizable if -4bc < (a-d)^2
is not diagonalizale if -4bc > (a-d)^2

2. Relevant equations

For an nxn matrix, if there are n distinct eigenvalues then the matrix is diagonalizable.
For an nxn matrix, if there are n linearly independent eigenvetors then the matrix is diagonalizable.

3. The attempt at a solution

Characteristic equation for eigenvalues:
|λ-a,-b;-c,λ-d| = λ^2 + λ(a-d) + (ab - bc) = 0
λ = 0.5 * (d - a plus-or-minus sqrt((a-d)^2 - 4ad + 4bc))

For part 1, I need to show that there are two distinct solutions to this quadratic equation (b^2 - 4ac > 0: a,b,c from general quadratic eq - not from this problem)
I know that (a-d)^2 + 4bc > 0

However, I do not know how to show that (a-d)^2 - 4ad + 4bc > 0

Last edited: Dec 12, 2011
2. Dec 12, 2011

### Staff: Mentor

Expand the (a - d)2 term and then combine like terms. You're almost there!

3. Dec 12, 2011

### I like Serena

You've got a minus sign wrong in your characteristic equation.

4. Dec 13, 2011

### Rockoz

I had trouble with the same exact problem recently. Your process is correct. You just have a minus sign error.