Monitor other linux stations through SSH, passwordless commands

The proxy server is with IP address:

10.1.10.16

Destination server to monitor is with IP:

10.1.10.17

Generate SSH key:

ssh-keygen -t rsa -b 2048

This will show:

Generating public/private rsa key pair.
Enter file in which to save the key (/root/.ssh/id_rsa):
Enter passphrase (empty for no passphrase):
Enter same passphrase again:
Your identification has been saved in /root/.ssh/id_rsa.
Your public key has been saved in /root/.ssh/id_rsa.pub.
The key fingerprint is:
SHA256:uqqJUWI2MGfgvXQlIdWX48FvJxvOlCukwYhGirDbCU0 root@zabbix.pro
The key's randomart image is:
+---[RSA 2048]----+
|.  ..+o.. .      |
|o.E.. o. *       |
|===o..o o + .    |
|==ooo. o o B .   |
|.Boo    S = *    |
|+o+    o . =     |
|.     .   .      |
| o .   .         |
|. o....          |
+----[SHA256]-----+

Copy public key to the home directory of destination station:

scp ~/.ssh/id_rsa.pub root@10.1.10.17:~

a password will be required to successfuly complete the command.

Now log in to destination station:

ssh root@10.1.10.17

A password will be required! Append the the SSH allowed_hosts file with the new public key:

cat id_rsa.pub >> ~/.ssh/authorized_keys

see how much keys are actually installed:

cat ~/.ssh/authorized_keys

Turn on SSH agent if it is not running:

ssh-agent

Exit remote station:

exit

try again authorize:

ssh root@10.1.10.17

This time the password will not be prompted.