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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