Server B need to access Server A  using SSH Public Key authentication (Does not need to have password login)

At Server B
To use public keys with an ssh server, you’ll first need to generate a public/private key pair at Server A:

$ ssh-keygen -t rsa
Generating public/private rsa key pair.
Enter file in which to save the key (/home/root/.ssh/id_rsa):
Enter passphrase (empty for no passphrase): 
Enter same passphrase again: 
Your identification has been saved in /home/root/.ssh/id_rsa.
Your public key has been saved in /home/root/.ssh/
The key fingerprint is:
a1:3c:a4:eb:18:95:1b:16:b1:a3:28:18:ga:81:3a:ba root@localhost

At Server B
To install the generated cert/key from Server B to Server A (or any server that ServerB need to access) with IP address :

$ ssh "mkdir .ssh; chmod 0700 .ssh"
$ scp .ssh/

By now, ServerB will be able to access ServerA without need to key in login password.

Alternative way of install cert from ServerB to ServerA

ssh-copy-id -i .ssh/ root@serverA