# Nullspace of matrix with Cosine

I have a question. I consider following matrix:

A = {{1, 0}, {0, Cos[b]}}


and i would like to find its nullspace. If b=Pi/2, rank of A is 1 and then I can calculate its nullspace. However, if I write matrix A in the above-mentioned symbolic form, Mathematica returns that nullspace does not exist. How can I solve this problem?

How can I ask to Mathematica, for what values of b nullspace of A is not empty?

Yeah, symbolic computations in Mathematica are usually performed for generic choices of values for the symbols. But Eigensystem can help here:
{eigenvalues,eigenvectors} = Eigensystem[A]

Now you can see: If Cos[b] == 0 then {0, 1} lies in the null space of A.
• The matrix A has rank 1 if precisely one element of eigenvalues is nonzero. In the case of a 2 by 2 matrix, Mathematica can be asked like this: Reduce[Or @@ Thread[eigenvalues == 0], b] – Henrik Schumacher Apr 24 '18 at 10:23