Matrix inversion

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Overview: Matrix inversion

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This article describes the inversion of matrices. It starts with a short explanation of what the inverse of a matrix actually is. Afterwards two different approaches to determine the inverse of a matrix are presented.

The inverse of an n-by-n square matrix $\mathbf{A}$ is denoted as $\mathbf{A}^{-1}$ and defined such that

$\mathbf{A}\mathbf{A}^{-1}=\mathbf{A}^{-1}\mathbf{A}=\mathbf{I}_n$

where $\mathbf{I}_n$ is the n-by-n identity matrix.
Prerequesite for the inversion is, that $\mathbf{A}$ is an n-by-n square matrix and that $\mathbf{A}$ is regular. Regular means that the row and column vectors are linearly independent and so the determinant is nonzero:

$det(\mathbf{A})\ne0$

Otherwise the matrix is called singular.

Before determining the inverse of a matrix it is always useful to compute the determinant and check whether the matrix is regular or singular. If it is singular it is not possible to determine the inverse because there is no inverse. The last two subarticles describe two of the common procedures to determine the inverse of a matrix.

1. Minors and cofactors
2. Determinant of a 4-by-4 matrix
3. Gauß-Jordan-Algorithm
4. Adjugate Formula

Example: inverse of matrix $\mathbf{A}_e = \left[\begin{array}{cccc} 1 & 2 & 0 & 0\\ 3 & 0 & 1 & 1\\ 0 & 1 & 0 & 0\\ 0 & 0 & 2 & 1 \end{array}\right] ,\quad \mathbf{A}_e^{-1} = \left[\begin{array}{cccc} 1 & 0 & -2 & 0\\ 0 & 0 & 1 & 0\\ 3 & -1 & -6 & 1\\ -6 & 2 & 12 & -1 \end{array}\right]$ $\begin{align} \mathbf{A}_e\mathbf{A}_e^{-1} &= \left[\begin{array}{cccc} 1 & 2 & 0 & 0\\ 3 & 0 & 1 & 1\\ 0 & 1 & 0 & 0\\ 0 & 0 & 2 & 1 \end{array}\right]\cdot \left[\begin{array}{cccc} 1 & 0 & -2 & 0\\ 0 & 0 & 1 & 0\\ 3 & -1 & -6 & 1\\ -6 & 2 & 12 & -1 \end{array}\right]\\&= \left[\begin{array}{cccc} 1+0+0+0 & 0+0+0+0 & -2+2+0+0 & 0+0+0+0\\ 3+0+3-6 & 0+0-1+2 & -6+0-6+12 & 0+0+1-1\\ 0+0+0+0 & 0+0+0+0 & 0+1+0+0 & 0+0+0+0\\ 0+0+6-6 & 0+0-2+2 & 0+0-12+12 & 0+0+2-1 \end{array}\right]\\&= \left[\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array}\right]= \mathbf{I}_n \end{align}$