What is 

Which of the following statements is correct? Surprisingly hard to get this right without your eyes glazing over :)

What is 

Let A be the matrix of a 3d rotation of 60 degrees in the counterclockwise direction around the vector (1,1,1)^T. How many eigenvalues does this rotation have?

Do you understand the following:

To be able to diagonalise an nxn matrix, it has to have n linearly independent eigenvectors.

For a matrix to be diagonalizable it does not necessarily have to have n different eigenvalues. 

If v₁, v₂, v₃, ... are linearly independent eigenvectors with corresponding eigenvalues  l₁, l₂, l₃, ... , respectively. To make up the matrix D we can add the eigenvalues in any order. However, when we then build the corresponding diagonalizing matrix T we have to use the corresponding eigenvectors in the same order.

For a matrix to be diagonalizable is a good thing :)

Let A be the matrix of a 3d reflection through a plane that contains the origin.  Any such matrix has the same two eigenvalues. What’s the smaller of these two eigenvalues?

The matrix can be diagonalised. What's the smallest non-zero entry of the diagonal matrix?

This matrix 

has eigenvalues 1 and 2. If 

is an eigenvector corresponding to the eigenvalue 1, what is a?

This is a matrix A in Mathematica format {{3, 1, 0, 0, 0}, {1, 3, 0, 0, 0}, {0, 0, 4, 1, 0}, {0, 0, 0, 4, 0}, {0, 0, 0, 0, 4}}.

This matrix 

has eigenvalues 1 and 2. If 

is an eigenvector corresponding to the eigenvalue 2, what is a?