# Homework Help: Proving eigenvalues?

1. Aug 20, 2011

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

Prove: If a, b, c, and d are integers such that a+b=c+d, then
A=
[a b]​
[c d] ​
has integer eigenvalues, namely,$λ_1{}$=a+b and $λ_2{}$=a-c

2. Relevant equations

No relevant equation.

3. The attempt at a solution

No idea :(

2. Aug 20, 2011

### fzero

Do you know how to compute the eigenvalues of a 2x2 matrix? Try that first before applying the additional information given in the problem.

3. Aug 20, 2011

### HallsofIvy

You don't really need to calculate the eigenvalues, you are only asked to show that a+b and a- c are eigenvalues- and to do that you use the definition of "eigenvalue".

That is, do there exist values, x and y, such that
$$\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= (a+b)\begin{bmatrix}u \\ v\end{bmatrix}$$
or values u and y such that
$$\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}u \\ v\end{bmatrix}= (a-b)\begin{bmatrix}u \\ v\end{bmatrix}$$
?

4. Aug 20, 2011

### I like Serena

Just for fun, here's yet another method.

The product of the eigenvalues is equal to the determinant.
The sum of the eigenvalues is equal to the trace.
In a quadratic equation the roots are uniquely identified by their product and their sum.
Use a+b=c+d to eliminate d in your equations.

5. Aug 20, 2011

### micromass

Hmm, that's cute