Eigenvalues

The matrix (2x2)

1 1
3 3


has eigenvalues of 0 and 4. What is a reason we should have known one of the eigenvalues of A??


AND

Eigenvectors of A are vectors x such that A fixes the direction of x. Give the argument that shows this is equivalent to x being in the nullspace of a certain matrix.

The two columns are the same. That means the matrix is not invertible (determinant is 0) and so one of the eigenvalues must be 0. The determinant of a matrix is the product of its eigenvalues. If the determinant is 0, then at least one of the eigenvalues must be 0.

Another way of looking at it is that the "column space" is spanned by <1, 3> and so is one-dimensional. That means that the nullspace has dimension 2- 1= 1> 0. There exist non-zero vectors v such that Av= 0= 0v so, again, 0 is an eigenvalue.

Totally makes sense! Thank you so much!