Please give the steps to find the inverse of a 2X2 and 3X3 matrix.

To calculate the inverse of ANY matrix A, we first need to find its determinant, |A|, and then we need to find its adjoint matrix, adj A. Then we can have the inverse matrix as:-

$A^{-1}=\frac{1}{\left|A\right|}\times adj A$

Now, if A is a 2x2 matrix, you can apply a shortcut because (i) it is a square matrix, (ii) because it is of order only 2.

Say $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

Then, $A^{-1}=\frac{1}{\left|A\right|}\times \left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]=\left({\displaystyle \frac{1}{ad-bc}}\right)\times \left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

There is no real shortcut to find the inverse of a 3x3 matrix. However, if you can remember it correctly, you can use the following:-

Say 


Then,