# Inverse matrix of 2-by-2 matrix, 3-by-3
matrix, 4-by-4 matrix

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## Inverse matrix of 2 $\times$ 2 matrix

There exists an inverse matrix of A when detA=ad-bc0, and it is

## Inverse matrix of 3 $\times$ 3 matrix

There exists an inverse matrix of A when

detA=a_{11}a_{22}a_{33}+a_{21}a_{32}a_{13}+a_{31}a_{12}a_{23}-a_{11}a_{32}a_{23}-a_{31}a_{22}a_{13}-a_{21}a_{12}a_{33}

0, and it is

## Inverse matrix of 4 $\times$ 4 matrix

If

then there exists an inverse matrix of A, and it is

where

## Inverse matrix of NxN matrix

From the analogy of the above formulae, the computation time
of inverse matrix of NxN matrix will be O(N^{3}N!).
Computing inverse matrix with Gauss-Jordan method, the method
using LU decomposition, and the method using SVD, will take a
computation time of O(N^{3}) (not confident). I will
recommend not to use the formula for calculating inverse matrix
of NxN matrix which N >= 4.

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