Inverse of $4 \times 4 $ matrix Would you be so kind to provide me some examples of  $4 \times 4 $ matrices that satisfy the relation  $AB = I $
I am completely clueless to find randomly any two matrix $A$ & $B$. Do we have any rules or shortcuts to find them or do we have to do a trial and error method till we get the answer. I hope that's not the right way.
Please shed some light towards this!

The standard "beginner's" method for matrix inversion is Gauss-Jordan elimination. 
Here are some $4 \times 4$ examples that work out reasonably simply.
$$A = \left[ \begin {array}{cccc} 2&-2&-1&-3\cr 2&-3&0&-3
\cr -1&-1&2&0\cr -1&3&-2&-2
\end {array} \right],\ B = A^{-1} =  
 \left[ \begin {array}{cccc} -12&11&-9/2&3/2\cr -10&9
&-7/2&3/2\cr -11&10&-7/2&3/2\cr 2&-2
&1/2&-1/2\end {array} \right]$$ 
$$A = \left[ \begin {array}{cccc} -1&0&1&0\\ 2&0&1&1
\\ -1&1&3&-1\\ 1&0&3&1\end {array}
 \right],\ B = A^{-1} = 
 \left[ \begin {array}{cccc} -2&-1&0&1\\ 6&6&1&-5
\\ -1&-1&0&1\\ 5&4&0&-3
\end {array} \right] 
$$
