Invert Matrix: Overview, Explanation & Example

This article is about the inverse matrix. In this chapter you will find out what this is all about, which terms and rules are important to you and how you can use them in examples. We can assign the chapter to the matrices and thus to the subject of math.

Inverse Matrix – What is it all about?

Before we get into what properties an inverse matrix has and how we can invert a matrix, let’s briefly review some basics.

reciprocal of a number

In mathematics we have already learned about powers and power rules for numbers or fractions. A fraction can be written as a number with a negative power, such as:

So this would be the one reciprocal the number 3. If we multiply a number by its reciprocal, the result is always 1.

From matrix to inverse matrix

The basis for an inverse matrix is ​​the matrix itself. We already know the different forms of matrices from the chapter on matrices. If several matrices are linked together, we have to deal with matrix calculus. If the basics of the matrices are unclear to you, please read the relevant chapter again.

The term «inverse» originally comes from Latin and means something like «vice versa». With an inverse matrix, the matrix is ​​also inverted and we get a inverse matrix.

Analogously to the normal numbers, an inverse matrix also receives a negative power. An inverse matrix is ​​denoted by the superscript -1.

  • Matrix A
  • inverse matrix

Below we show you an example of a matrix A and its inverse matrix. For the sake of simplicity, we only use a 2×2 matrix at first.

Multiplying the matrix A by the inverse matrix we get an identity matrix.

We will explain later how the inverse matrix of an original matrix A can be calculated. First we deal with the properties and calculation rules of the inverse matrices.

Existence of the inverse matrix

Not every matrix can be reversed or inverted. Certain conditions must be met in order for an inverse matrix to be calculated. A matrix is ​​invertible if:

  • The matrix A is square.
  • The determinant of the matrix is ​​non-zero.

As an example we take the following matrices A and B. We want to check whether the prerequisites are fulfilled and whether inverse matrices exist for these matrices.

For the matrix A, the first condition is already not fulfilled because the matrix is ​​not square. With this we can already deny the question of invertibility. In contrast, the matrix B is square with two rows and two columns and thus satisfies the first requirement. With the calculation of the determinant, the second requirement is now checked.

Consequently, for the matrix B there exists an inverse matrix. However, not every square matrix has an inverse matrix, so both requirements must be checked. Depending on whether a matrix is ​​invertible or non-invertible, it can be named differently:

  • Invertible Matrix -> regular matrix
  • Non-invertible matrix -> singular matrix

Calculation rules for inverse matrices

We already know when a matrix is ​​invertible. However, there are some important properties and rules to keep in mind when dealing with inverse matrices. You should already be familiar with the basic calculation rules for matrices from matrix calculation.

  • invert an inverse matrix:

By inverting an already inverted matrix we get the original matrix A again. It follows:

  • multiplication of inverse matrices:

Inverting a matrix product is equivalent to the product of the respective inverse. However, the order of the matrices must be observed when multiplying.

  • multiplication with scalars:

Inverse matrices can also be multiplied by scalars. Here, the reciprocal of the scalar is multiplied. This follows:

  • invert a transposed matrix:

Inverting a transposed matrix is ​​equivalent to transposing an inverse matrix. The following applies:

For an orthogonal matrix, the inverse matrix is ​​the same as the transposed matrix:

invert matrices

In principle, various methods can be used to calculate an inverse matrix, such as:

  • Gauss-Jordan algorithm
  • adjuncts
  • Cramer’s rule

Below we show the inversion of a matrix A using the Gauss-Jordan method using a simple example.

We already know that multiplying the matrix A by its inverse matrix A-1 must result in the identity matrix E. This information is used for the calculation.

1. Form block matrix (A|E).

First, a combined block matrix is ​​formed from the matrix A and the identity matrix E. For easier understanding, the brackets are also omitted.

The matrix A is on the left-hand side and the identity matrix E on the right-hand side. The goal is to use the Gauss-Jordan method to transform the matrices in such a way that the identity matrix is ​​generated on the left-hand side.

2. Forming 1

A first possible transformation would be to multiply the second line by a factor of 3.

With this we get:

3. reshaping 2

Next we subtract row 1 from row 2.

The result is:

4. reshaping 3

Now we can subtract from row 1 5 times row 2.

Thus we get:

5. reshaping 4

By dividing row 1 we get the last transformation.

The result is:

As we can see, the rows have been reshaped so that we get an identity matrix on the left. If this is the case, the inverse matrix can be read from the right-hand side.

6. Read inverse matrix

The inverse matrix is ​​then:

Thus, a matrix can be inverted from a matrix using the Gauss-Jordan algorithm. We have now learned all the important basics about inverse matrices. You can use the following exercise example to test your knowledge on this topic.

Inverse matrix – practice example

task: Using the given matrix A, show that it is invertible.

Solution: Whether a matrix is ​​invertible depends on whether the requirements for invertibility are met.

  • The matrix must be square.
  • The determinant of the matrix is ​​non-zero.

The matrix A has 3 columns and 3 rows, is square and fulfills the first condition. In addition, we calculate the determinant of the matrix A.

The determinant is therefore non-zero. This also fulfills the second requirement and the matrix can be inverted.

Below you will find a short overview with the most important information.

Invert matrix – everything important at a glance

  • An inverse matrix is ​​also called inverse matrix designated.
  • The inverse matrix is ​​given by the spelling Marked A-1.
  • Multiplying a matrix by its inverse matrix gives an identity matrix:
  • A matrix is ​​only invertible if the following conditions are met:
    1. The Matrix is square.
    2. The determinant of the matrix is non-zero.
  • Invertible matrices are also called regular named matrices.
  • singular Matrices are non-invertible matrices.
  • The following calculation rules must be observed in the calculation:
  • Matrices can be inverted by different methods:
    1. Gauss-Jordan algorithm
    2. adjuncts
    3. Cramer’s rule
  • The block matrix (E|A-1) can be formed from the block matrix (A|E) by conversion, and the inverse matrix can thus be read.