Proving Integer Eigenvalues of Matrix A

[hadizainud]

Homework Statement

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

Homework Equations

No relevant equation.

The Attempt at a Solution

No idea :(

 

[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.

 

[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
$$\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}u \\ v\end{bmatrix}= (a-b)\begin{bmatrix}u \\ v\end{bmatrix}$$
?

 

[I like Serena]

Welcome to PF, hadizainud!

The methods of fzero and HallsofIvy will bring you your answer.
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.

 

[micromass]

Welcome to PF, hadizainud!

The methods of fzero and HallsofIvy will bring you your answer.
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.

Hmm, that's cute