# Center of $SU(3)$

I assumed a 3x3 matrix of the form $$A= \begin{pmatrix} a & b & c\\ d & e & f\\ k & l & m \end{pmatrix}$$ Then, since we know that the center is always an Abelian invariant subgroup and AB=BA, and the Gell-Mann matrices belong in SU(3), i took $$B=\begin{pmatrix} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}$$ and $$B=\begin{pmatrix} 1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 0 \end{pmatrix}$$ which made the first matrix $$A=\begin{pmatrix} a & 0 & 0\\ 0 & a & 0\\ 0 & 0 & m \end{pmatrix}$$ Know that $$det(A) =1$$ led to $$a^2m=1$$ And that's my question. Shouldn't i get numbers and not a and m for the center of the group?

The Gell-Mann matrices do not belong to the group $${\rm SU}(3)$$. They are a vector-space basis for $${\rm Lie}\{{\rm SU}(3)\}$$, the Lie algebra associated with the group. The centre $$Z\subset {\rm SU}(3)$$ is the set of unit-determinant unitary matrices that commute with all elements of $${\rm SU}(3)$$. Schur's lemma tells us that they have to be multiples of the identity matrix $${\mathbb I}$$. So the question for you: If $$z\,{\mathbb I} \in {\rm SU}(3)$$ what can we say about possible values of $$z$$?