Matrix inversion

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Overview: Matrix inversion

Next: Minors and cofactors →

This article describes the inversion of matrices. It starts with a short explanation of what the inverse of a matrix actually is. Afterwards subarticles present some matrix features and two different approaches to determine the inverse of a matrix based on these features.

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. For 3-by-3 and smaller matrices there are simple formulas to compute the determinant. Useful to compute the determinant of larger matrices are the minors and cofactors which are explained in the first subarticle. Based on these an example formula to compute the determinant of a 4-by-4 matrix is presented afterwards. 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 $\mathbf{A}_e = \left[\begin{array}{cccc} 1 & 2 & 0 & 0\\ 3 & 0 & 1 & 1\\ 0 & 1 & 0 & 0\\ 0 & 0 & 2 & 1 \end{array}\right] ,\quad \mathbf{A}_e^{-1} = \left[\begin{array}{cccc} 1 & 0 & -2 & 0\\ 0 & 0 & 1 & 0\\ 3 & -1 & -6 & 1\\ -6 & 2 & 12 & -1 \end{array}\right]$ $\begin{align} \mathbf{A}_e\mathbf{A}_e^{-1} &= \left[\begin{array}{cccc} 1 & 2 & 0 & 0\\ 3 & 0 & 1 & 1\\ 0 & 1 & 0 & 0\\ 0 & 0 & 2 & 1 \end{array}\right]\cdot \left[\begin{array}{cccc} 1 & 0 & -2 & 0\\ 0 & 0 & 1 & 0\\ 3 & -1 & -6 & 1\\ -6 & 2 & 12 & -1 \end{array}\right]\\&= \left[\begin{array}{cccc} 1+0+0+0 & 0+0+0+0 & -2+2+0+0 & 0+0+0+0\\ 3+0+3-6 & 0+0-1+2 & -6+0-6+12 & 0+0+1-1\\ 0+0+0+0 & 0+0+0+0 & 0+1+0+0 & 0+0+0+0\\ 0+0+6-6 & 0+0-2+2 & 0+0-12+12 & 0+0+2-1 \end{array}\right]\\&= \left[\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array}\right]= \mathbf{I}_n \end{align}$