  Let $\mathcal{H}$ be a complex Hilbert space and let $A$ be a positive
operator on $\mathcal{H}$. We obtain new bounds for the $A$-numerical radius of
operators in semi-Hilbertian space $\mathcal{B}_A(\mathcal{H})$ that generalize
and improve on the existing ones. Further, we estimate bounds for the
$B$-operator seminorm and $B$-numerical radius of $2\times 2$ operator
matrices, where $B=\mbox{diag}(A,A)$. The bounds obtained here improve on the
existing ones.
