The inverse of a matrix may be found using several different methods. The method that is guaranteed to work is by augmenting a n×n matrix with
I
n
{\displaystyle I_{n}}
[
1
3
2
2
]
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[
1
0
0
1
]
{\displaystyle {\begin{bmatrix}1&3\\2&2\end{bmatrix}}{\Bigg |}{\begin{bmatrix}1&0\\0&1\end{bmatrix}}}
.
.
.
{\displaystyle ...}
[
1
0
0
1
]
|
[
−
1
2
3
4
1
2
−
1
4
]
{\displaystyle {\begin{bmatrix}1&0\\0&1\end{bmatrix}}{\Bigg |}{\begin{bmatrix}-{\cfrac {1}{2}}&{\cfrac {3}{4}}\\{\cfrac {1}{2}}&-{\cfrac {1}{4}}\end{bmatrix}}}
The inverse of the matrix is the second augmented matrix. In this case,
[
−
1
2
3
4
1
2
−
1
4
]
{\displaystyle {\begin{bmatrix}-{\cfrac {1}{2}}&{\cfrac {3}{4}}\\{\cfrac {1}{2}}&-{\cfrac {1}{4}}\end{bmatrix}}}
