<s>[INST] Inverse of a 2x2 matrix with an example Need to find inverse of this matrix:
$
\begin {bmatrix}
1 & 3/5\\
0 & 1\\
\end {bmatrix}
$
This is how it has been solved:
$
\begin {bmatrix}
1 & 3/5\\
0 & 1\\
\end {bmatrix}
$ $
\begin {bmatrix}
x_1 & x_2\\
x_3 & x_4\\
\end {bmatrix}
$ = $
\begin {bmatrix}
1 & 0\\
0 & 1\\
\end {bmatrix}
$ - step 1
$
\begin {bmatrix}
1 & 0\\
0 & 1\\
\end {bmatrix}
$ $
\begin {bmatrix}
x_1 & x_2\\
x_3 & x_4\\
\end {bmatrix}
$ = $
\begin {bmatrix}
1 & -3/5\\
0 & 1\\
\end {bmatrix}
$ - step 2
From step 1 to step 2, -3/5 added to first row, second column of $
\begin {bmatrix}
1 & -3/5\\
0 & 1\\
\end {bmatrix}
$
Also, -3/5 to first row,second column of $
\begin {bmatrix}
1 & 0\\
0 & 1\\
\end {bmatrix}
$
My query is if it is correct to add or deduct a particular number from say first column, first row on the right side and on the left side and both leave the equation intact?
 [/INST] One may find the inverse matrix by row operations on the augmented matrix:
$$
  \left[\begin{array}{rr|rr}
    1 & 3/5  & 1 &0 \\
    0 & 1  & 0 &1 \\
  \end{array}\right]\overset{R_1'=-3/5R2+R1}{\longrightarrow} 
  \left[\begin{array}{rr|rr}
    1 & 0  & 1 &-3/5 \\
    0 & 1  & 0 &1 \\
  \end{array}\right],
$$
so $$
\begin {bmatrix}
1 & 3/5\\
0 & 1\\
\end {bmatrix}^{-1}
=
\begin {bmatrix}
1 & -3/5\\
0 & 1\\
\end {bmatrix}.
$$
It is also useful to remember the general expression for the inverse of a 2x2 matrix:
$$
\begin {bmatrix}
a & b\\
c & d\\
\end {bmatrix}^{-1}=\frac{1}{ad-bc}
\begin {bmatrix}
d & -b\\
-c & a\\
\end {bmatrix}
$$
if the determinant $ad-bc\neq 0.$
</s>