I have to find 4 things for the Matrix A which is a 3x3 matrix with all values equal to 1

A=

1 1 1

1 1 1

1 1 1

a.) Without having to make extensive calculations explain why 0 is an eigenvalue of A

b.) Find 2 linearly independent Eigenvectors for the Eigenvalue 0

c.) The e-value 0 has both geometric and algebraic multiplicity 2. Find the remaining eigenvalue and associated eigenvector without extensive calculations.

d.) Determine the matrices S and Λ and show A S = S Λ

I've uploaded a picture of my work since I'm not too familiar with it all up.

a.) I know that there are 2 eigenvalues (0 and 3) because of my calculations above but I'm not sure how to explain that without extensive calculations. I think it has to do with the trace?

b.) I also know that two linearly independent eigenvectors are v1 = (-1,0,1) and v2 = (-1,1,0) but I'm not sure how to show this.

c.) I think the remaining Eigenvalue is just 3, but I knew that from before hand when I found the e-values the long way. And it's associated e-vector would just be (1,1,1) I'm not sure how to show this in a non extensive calculations way.

and as for d.) I don't really know

Thanks in advance