# Adjugate Formula

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The adjugate formula defines the inverse of an n-by-n square matrix as

where is the so called **adjugate matrix** of . The adjugate matrix is the transposed of the cofactor matrix:

And the **cofactor matrix** is just a matrix where each cell corresponds to the related cofactor:

So to determine the inverse of an n-by-n square matrix you have to compute the n square cofactors, then transpose the resulting cofactor matrix and divide all the values by the determinant.

Example: inverse of a 2-by-2 matrix using the adjugate formula
The following matrix is already used in examples for the minors and cofactors and for the determinant.
The computation of the cofactors is easy in this case because the minors correspond to the determinants of 1-by-1 matrices, that are the value itself. So the cofactor matrix of matrix is The transposed of the cofactor matrix which corresponds to the adjugate matrix then appears as
The determinant is computed in the example for a determinant of a matrix and equals 1. So the inverse of is determined as For a proof please have a look at the example in the main article of matrix inversion. |

Example: inverse of a 3-by-3 matrix using the adjugate formula
The following matrix is already used in examples for the minors and cofactors and for the determinant.
The calculation of two of the cofactors of has already been described in the example for minors and cofactors. The resulting cofactor matrix of matrix is The transposed of the cofactor matrix which corresponds to the adjugate matrix then appears as
The determinant is computed in the example for a determinant of a matrix and equals 1. So the inverse of is determined as For a proof please have a look at the example in the main article of matrix inversion. |

Example: inverse of a 4-by-4 matrix using the adjugate formula
In this example again the transformation matrix that is introduced in the robotics script in chapter 3 on page 3-37 is used:
The calculation of two of the cofactors of has already been described in the example for minors and cofactors. The resulting cofactor matrix of matrix is The transposed of the cofactor matrix which corresponds to the adjugate matrix then appears as
The determinant is computed in the example for a determinant of a matrix and equals 1. So the inverse of is determined as For a proof please have a look at the example in the main article of matrix inversion. |