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

Inverse matrices of 2-by-2 matrix, 3-by-3 matrix, 4-by-4 matrix are shown here.

Inverse matrix of 2 $\times$ 2 matrix

$A = (\begin{matrix} a & b \\ c & d \end{matrix})$

There exists an inverse matrix of $A$ when $d e t A = a d - b c \neq 0$ , and it is

$A^{- 1} = \frac{1}{a d - b c} (\begin{matrix} d & - b \\ - c & a \end{matrix})$

Inverse matrix of 3 $\times$ 3 matrix

$A = (\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix})$

There exists an inverse matrix of $A$ when
$d e t A = a_{11} a_{22} a_{33} + a_{12} a_{23} a_{31} + a_{13} a_{21} a_{32} - a_{13} a_{22} a_{31} - a_{11} a_{23} a_{32} - a_{12} a_{21} a_{33}$
$\neq 0$ , and it is

$A^{- 1} = \frac{1}{d e t A} (\begin{matrix} a_{22} a_{33} - a_{23} a_{32} & a_{13} a_{32} - a_{12} a_{33} & a_{12} a_{23} - a_{13} a_{22} \\ a_{23} a_{31} - a_{21} a_{33} & a_{11} a_{33} - a_{13} a_{31} & a_{13} a_{21} - a_{11} a_{23} \\ a_{21} a_{32} - a_{22} a_{31} & a_{12} a_{31} - a_{11} a_{32} & a_{11} a_{22} - a_{12} a_{21} \end{matrix})$

Inverse matrix of 4 $\times$ 4 matrix

$A = (\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{matrix})$

$d e t A$
$= a_{11} a_{22} a_{33} a_{44} + a_{11} a_{23} a_{34} a_{42} + a_{11} a_{24} a_{32} a_{43}$
$+ a_{12} a_{21} a_{34} a_{43} + a_{12} a_{23} a_{31} a_{44} + a_{12} a_{24} a_{33} a_{41}$
$+ a_{13} a_{21} a_{32} a_{44} + a_{13} a_{22} a_{34} a_{41} + a_{13} a_{24} a_{31} a_{42}$
$+ a_{14} a_{21} a_{33} a_{42} + a_{14} a_{22} a_{31} a_{43} + a_{14} a_{23} a_{32} a_{41}$
$- a_{11} a_{22} a_{34} a_{43} - a_{11} a_{23} a_{32} a_{44} - a_{11} a_{24} a_{33} a_{42}$
$- a_{12} a_{21} a_{33} a_{44} - a_{12} a_{23} a_{34} a_{41} - a_{12} a_{24} a_{31} a_{43}$
$- a_{13} a_{21} a_{34} a_{42} - a_{13} a_{22} a_{31} a_{44} - a_{13} a_{24} a_{32} a_{41}$
$- a_{14} a_{21} a_{32} a_{43} - a_{14} a_{22} a_{33} a_{41} - a_{14} a_{23} a_{31} a_{42}$
$\neq 0$

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

$A^{- 1} = \frac{1}{d e t A} (\begin{matrix} b_{11} & b_{12} & b_{13} & b_{14} \\ b_{21} & b_{22} & b_{23} & b_{24} \\ b_{31} & b_{32} & b_{33} & b_{34} \\ b_{41} & b_{42} & b_{43} & b_{44} \end{matrix})$

where

$b_{11} = a_{22} a_{33} a_{44} + a_{23} a_{34} a_{42} + a_{24} a_{32} a_{43} - a_{22} a_{34} a_{43} - a_{23} a_{32} a_{44} - a_{24} a_{33} a_{42}$
$b_{12} = a_{12} a_{34} a_{43} + a_{13} a_{32} a_{44} + a_{14} a_{33} a_{42} - a_{12} a_{33} a_{44} - a_{13} a_{34} a_{42} - a_{14} a_{32} a_{43}$
$b_{13} = a_{12} a_{23} a_{44} + a_{13} a_{24} a_{42} + a_{14} a_{22} a_{43} - a_{12} a_{24} a_{43} - a_{13} a_{22} a_{44} - a_{14} a_{23} a_{42}$
$b_{14} = a_{12} a_{24} a_{33} + a_{13} a_{22} a_{34} + a_{14} a_{23} a_{32} - a_{12} a_{23} a_{34} - a_{13} a_{24} a_{32} - a_{14} a_{22} a_{33}$
$b_{21} = a_{21} a_{34} a_{43} + a_{23} a_{31} a_{44} + a_{24} a_{33} a_{41} - a_{21} a_{33} a_{44} - a_{23} a_{34} a_{41} - a_{24} a_{31} a_{43}$
$b_{22} = a_{11} a_{33} a_{44} + a_{13} a_{34} a_{41} + a_{14} a_{31} a_{43} - a_{11} a_{34} a_{43} - a_{13} a_{31} a_{44} - a_{14} a_{33} a_{41}$
$b_{23} = a_{11} a_{24} a_{43} + a_{13} a_{21} a_{44} + a_{14} a_{23} a_{41} - a_{11} a_{23} a_{44} - a_{13} a_{24} a_{41} - a_{14} a_{21} a_{43}$
$b_{24} = a_{11} a_{23} a_{34} + a_{13} a_{24} a_{31} + a_{14} a_{21} a_{33} - a_{11} a_{24} a_{33} - a_{13} a_{21} a_{34} - a_{14} a_{23} a_{31}$
$b_{31} = a_{21} a_{32} a_{44} + a_{22} a_{34} a_{41} + a_{24} a_{31} a_{42} - a_{21} a_{34} a_{42} - a_{22} a_{31} a_{44} - a_{24} a_{32} a_{41}$
$b_{32} = a_{11} a_{34} a_{42} + a_{12} a_{31} a_{44} + a_{14} a_{32} a_{41} - a_{11} a_{32} a_{44} - a_{12} a_{34} a_{41} - a_{14} a_{31} a_{42}$
$b_{33} = a_{11} a_{22} a_{44} + a_{12} a_{24} a_{41} + a_{14} a_{21} a_{42} - a_{11} a_{24} a_{42} - a_{12} a_{21} a_{44} - a_{14} a_{22} a_{41}$
$b_{34} = a_{11} a_{24} a_{32} + a_{12} a_{21} a_{34} + a_{14} a_{22} a_{31} - a_{11} a_{22} a_{34} - a_{12} a_{24} a_{31} - a_{14} a_{21} a_{32}$
$b_{41} = a_{21} a_{33} a_{42} + a_{22} a_{31} a_{43} + a_{23} a_{32} a_{41} - a_{21} a_{32} a_{43} - a_{22} a_{33} a_{41} - a_{23} a_{31} a_{42}$
$b_{42} = a_{11} a_{32} a_{43} + a_{12} a_{33} a_{41} + a_{13} a_{31} a_{42} - a_{11} a_{33} a_{42} - a_{12} a_{31} a_{43} - a_{13} a_{32} a_{41}$
$b_{43} = a_{11} a_{23} a_{42} + a_{12} a_{21} a_{43} + a_{13} a_{22} a_{41} - a_{11} a_{22} a_{43} - a_{12} a_{23} a_{41} - a_{13} a_{21} a_{42}$
$b_{44} = a_{11} a_{22} a_{33} + a_{12} a_{23} a_{31} + a_{13} a_{21} a_{32} - a_{11} a_{23} a_{32} - a_{12} a_{21} a_{33} - a_{13} a_{22} a_{31}$

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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