Help with Linear Algebra Eigenvalues and Eigenvectors

[Sherri.kramer2@gmail.com]

I have a problem that I am not even really sure where to start.
Make a conjecture about the eigenvectors and eigenvalues of the matrix A corresponding to the given transformation by considering the geometric properties of multiplication by A. Confirm each of your conjectures with computations. a) Reflection about the line y=x.

I know the standard matrix for a reflection is <0 1> <1 0> and I know the lambda is the det(lambda I - A) but that's all I know. Can somebody please help me?

 

[stapel]

I have a problem that I am not even really sure where to start.

Make a conjecture about the eigenvectors and eigenvalues of the matrix A corresponding to the given transformation by considering the geometric properties of multiplication by A. Confirm each of your conjectures with computations. a) Reflection about the line y=x.

A good place to start might be with "considering the geometric properties of" "reflecting" a curve (or point, or whatever) "about the line y = x". What does the curve (or point, or whatever) do when you do this reflection? What does it look like? Etc.