# 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=a11a22a33+a21a32a13+a31a12a23-a11a32a23-a31a22a13-a21a12a33
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(N3N!). Computing inverse matrix with Gauss-Jordan method, the method using LU decomposition, and the method using SVD, will take a computation time of O(N3) (not confident). I will recommend not to use the formula for calculating inverse matrix of NxN matrix which N >= 4.

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