TITLE: shortcut for finding a inverse  of matrix
QUESTION [20 upvotes]: I need tricks or shortcuts to find the inverse of  $2 \times 2$ and $3 \times 3$ matrices. I have to take a time-based exam, in which I have to find the inverse of square matrices.

REPLY [6 votes]: So we want to find out a way to compute $2 \times 2 ~\text{ or }~ 3 \times 3$ matrix systems the most efficient way. Well I think the route that we want to go would be to use Cramer's Rule for the $2 \times 2 \text{ or } 3 \times 3$ case. To state the $2 \times 2$ case we will use the following:
For some coefficient matrix A=
$\left[ \begin{array}{rr} 
a & b \\
c & d 
\end{array} \right]$
$A^{-1}=\dfrac{1}{ad-bc} \cdot 

\left[ \begin{array}{rr} 
d & -b \\
-c & a 
\end{array} \right]~ \iff ad-bc \ne 0$  $~~~~~~~~~\Big($i.e., Det(A)$~\ne ~ 0\Big)$
For the $3 \times 3$ case, we will denote that as the folllowing:
$x_{1} = \dfrac{|b~~x_{2}~~x_{3}|}{|\bf{A}|}~~,$
$x_{2} = \dfrac{|x_{1}~~b~~x_{3}|}{|\bf{A}|}~~,$
$x_{3} = \dfrac{|x_{1}~~x_{2}~~b|}{|\bf{A}|}.$
This comes from the matrix equation: ${\bf{A\vec{x}}}={\bf{\vec{b}}},~~~$ where $\vec{x}=[x_{1}~~x_{2}~~x_{3}]^{T}$.
For the elements of matrix $A =  
\left|\begin{array}{rrr} 
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array} \right|,~~$
it can be extended for the solutions $x_{1},~x_{2},~x_{3}$ 
as so knowing that ${\bf|{A}| =} ~ |a_{ij}| ~ \not= ~ 0.$
$x_{1} = \dfrac{1}{|{\bf{A}}|}
\left|\begin{array}{rrr} 
b_1 & a_{12} & a_{13} \\
b_2 & a_{22} & a_{23} \\
b_3 & a_{32} & a_{33}
\end{array} \right|$,
$x_{2} = \dfrac{1}{|{\bf{A}}|}
\left|\begin{array}{rrr} 
a_{11} & b_1 & a_{13} \\
a_{21} & b_2 & a_{23} \\
a_{31} & b_3 & a_{33}
\end{array} \right|$,
$x_{3} = \dfrac{1}{|{\bf{A}}|}
\left|\begin{array}{rrr} 
a_{11} & a_{12} & b_1 \\
a_{21} & a_{22} & b_2 \\
a_{31} & a_{32} & b_3
\end{array} \right|$.
An alternate way of doing this would be using row reducing methods, known as either Gaussian Elimination( ref ) or Gauss-Jordan Elimination( rref ). 
I hope this helped out. Let me know if there if anything you do not understand.
Thanks.
Good Luck.