Kernel engineering, valid kernels, multipliying by constant =0?

I am reading Bishop, Pattern Recognition and Machine Learning. In the chapter about kernels rules are given for constructing kernels from existing valid kernels.

The first one being let k(x,y) be a valid kernel and c>0 a constant then

l(x,y)=c*k(x,y) is a valid kernel


I was wondering why does c>0? and not c>=0, since given the Gram matrix of K multiplying each entry by 0 will give us L. L is p.s.d since det(L)>=0, so it should be a valid kernel? (it seems like the Gram Matrix being p.s.d is a sufficient condition for a kernel to be valid) Where does this argument go wrong?