Difference between revisions of "Adjugate Formula"
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− | {{Navigation|before=[[Gauß-Jordan-Algorithm]]|overview=[[Matrix inversion]]|next=[[ | + | {{Navigation|before=[[Gauß-Jordan-Algorithm]]|overview=[[Matrix inversion]]|next=[[Transformations]]}} |
− | The adjugate formula defines the inverse of an n-by-n square matrix <math>\mathbf{A}</math> as | + | The adjugate formula defines the inverse of an n-by-n square matrix <math>\mathbf{A}</math> as |
− | <math>\mathbf{A}^{-1}=\frac{1}{\det(\mathbf{A})}\text{adj}(\mathbf{A})</math | + | :<math>\mathbf{A}^{-1}=\frac{1}{\det(\mathbf{A})}\text{adj}(\mathbf{A})</math> |
− | where <math>\text{adj}(\mathbf{A})</math> is the so called '''adjugate matrix''' of <math>\mathbf{A}</math>. The adjugate matrix is the transposed of the cofactor matrix: | + | where <math>\text{adj}(\mathbf{A})</math> is the so called '''adjugate matrix''' of <math>\mathbf{A}</math>. The adjugate matrix is the transposed of the cofactor matrix: |
− | <math> | + | :<math> |
\text{adj}(\mathbf{A})=\mathbf{C}(\mathbf{A})^T | \text{adj}(\mathbf{A})=\mathbf{C}(\mathbf{A})^T | ||
− | </math | + | </math> |
− | And the '''cofactor matrix''' <math>\mathbf{C}(\mathbf{A})</math> is just a matrix where each cell corresponds to the related [[Minors_and_cofactors|cofactor]]: | + | And the '''cofactor matrix''' <math>\mathbf{C}(\mathbf{A})</math> is just a matrix where each cell corresponds to the related [[Minors_and_cofactors|cofactor]]: |
− | <math> | + | :<math> |
\mathbf{C}(\mathbf{A})=\left[\begin{array}{cccc} | \mathbf{C}(\mathbf{A})=\left[\begin{array}{cccc} | ||
C_{11}(\mathbf{A}) & C_{12}(\mathbf{A}) & \cdots & C_{1n}(\mathbf{A})\\ | C_{11}(\mathbf{A}) & C_{12}(\mathbf{A}) & \cdots & C_{1n}(\mathbf{A})\\ | ||
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C_{n1}(\mathbf{A}) & C_{n2}(\mathbf{A}) & \cdots & C_{nn}(\mathbf{A}) | C_{n1}(\mathbf{A}) & C_{n2}(\mathbf{A}) & \cdots & C_{nn}(\mathbf{A}) | ||
\end{array}\right] | \end{array}\right] | ||
− | </math | + | </math> |
− | 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.<br/><br/> | + | 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 | ||
+ | |Title=inverse of a 2-by-2 matrix using the adjugate formula | ||
+ | |Contents= | ||
+ | The following matrix is already used in examples for the [[Minors and cofactors|minors and cofactors]] and for the [[Determinant of a matrix|determinant]]. | ||
+ | <br/> | ||
+ | :<math> | ||
+ | \mathbf{A}_2 = | ||
+ | \left[\begin{array}{cc} | ||
+ | 2&3\\ | ||
+ | 1&2 | ||
+ | \end{array}\right]</math> | ||
+ | |||
+ | 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 <math>\mathbf{A}_3</math> is<br/> | ||
+ | :<math> | ||
+ | \mathbf{C}(\mathbf{A}_2)= | ||
+ | \left[\begin{array}{cc} | ||
+ | (-1)^{1+1}\det | ||
+ | \left[\begin{array}{cc} | ||
+ | \Box&\Box\\ | ||
+ | \Box&2 | ||
+ | \end{array}\right] | ||
+ | & (-1)^{1+2}\det | ||
+ | \left[\begin{array}{cc} | ||
+ | \Box&\Box\\ | ||
+ | 1&\Box | ||
+ | \end{array}\right] | ||
+ | \\ \\ | ||
+ | (-1)^{2+1}\det | ||
+ | \left[\begin{array}{cc} | ||
+ | \Box&3\\ | ||
+ | \Box&\Box | ||
+ | \end{array}\right] | ||
+ | & (-1)^{2+2}\det | ||
+ | \left[\begin{array}{cc} | ||
+ | 2&\Box\\ | ||
+ | \Box&\Box | ||
+ | \end{array}\right] | ||
+ | \end{array}\right]= | ||
+ | \left[\begin{array}{cc} | ||
+ | (-1)^{1+1}\det[2] & (-1)^{1+2}\det[1]\\ | ||
+ | (-1)^{2+1}\det[3] & (-1)^{2+2}\det[2] | ||
+ | \end{array}\right]= | ||
+ | \left[\begin{array}{cc} | ||
+ | 2 & -1\\ | ||
+ | -3 & 2 | ||
+ | \end{array}\right] | ||
+ | </math> | ||
+ | |||
+ | The transposed of the cofactor matrix which corresponds to the adjugate matrix then appears as | ||
+ | <br/> | ||
+ | :<math>\mathbf{C}(\mathbf{A}_2)^T= | ||
+ | \left[\begin{array}{cc} | ||
+ | 2&-3\\ | ||
+ | -1&2 | ||
+ | \end{array}\right]=\text{adj}(\mathbf{A}_2)</math> | ||
+ | |||
+ | The determinant is computed in the example for a [[Determinant_of_a_matrix|determinant of a matrix]] and equals 1. So the inverse of <math>\mathbf{A}_2</math> is determined as <br/> | ||
+ | :<math> | ||
+ | \mathbf{A}_2^{-1}=\frac{1}{\det(\mathbf{A}_2)}\text{adj}(\mathbf{A}_2) | ||
+ | =\frac{1}{1} | ||
+ | \left[\begin{array}{cc} | ||
+ | 2&-3\\ | ||
+ | -1&2 | ||
+ | \end{array}\right] | ||
+ | = | ||
+ | \left[\begin{array}{cc} | ||
+ | 2&-3\\ | ||
+ | -1&2 | ||
+ | \end{array}\right] | ||
+ | </math> | ||
+ | For a proof please have a look at the example in the main article of [[Matrix inversion|matrix inversion]]. | ||
+ | }} | ||
{{Example | {{Example | ||
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\end{array}\right] | \end{array}\right] | ||
</math> | </math> | ||
+ | For a proof please have a look at the example in the main article of [[Matrix inversion|matrix inversion]]. | ||
}} | }} | ||
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0 & 0 & 0 & 1 | 0 & 0 & 0 & 1 | ||
\end{array}\right]</math> | \end{array}\right]</math> | ||
− | |||
The calculation of two of the cofactors of <math>^R\mathbf{T}_N</math> has already been described in the example for [[Minors_and_cofactors|minors and cofactors]]. The resulting cofactor matrix of matrix <math>^R\mathbf{T}_N</math> is<br/> | The calculation of two of the cofactors of <math>^R\mathbf{T}_N</math> has already been described in the example for [[Minors_and_cofactors|minors and cofactors]]. The resulting cofactor matrix of matrix <math>^R\mathbf{T}_N</math> is<br/> | ||
:<math> | :<math> | ||
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0&-2a&0&1 | 0&-2a&0&1 | ||
\end{array}\right]</math> | \end{array}\right]</math> | ||
− | |||
The transposed of the cofactor matrix which corresponds to the adjugate matrix then appears as | The transposed of the cofactor matrix which corresponds to the adjugate matrix then appears as | ||
<br/> | <br/> | ||
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0&0&0&1 | 0&0&0&1 | ||
\end{array}\right]=\text{adj}(^R\mathbf{T}_N)</math><br/> | \end{array}\right]=\text{adj}(^R\mathbf{T}_N)</math><br/> | ||
− | The determinant is computed in the example for a [[Determinant_of_a_matrix|determinant of a matrix]] and equals 1. So the inverse of <math>\mathbf{ | + | The determinant is computed in the example for a [[Determinant_of_a_matrix|determinant of a matrix]] and equals 1. So the inverse of <math>^R\mathbf{T}_N</math> is determined as <br/> |
:<math> | :<math> | ||
^R\mathbf{T}_N^{-1}=\frac{1}{\det(^R\mathbf{T}_N)}\text{adj}(^R\mathbf{T}_N) | ^R\mathbf{T}_N^{-1}=\frac{1}{\det(^R\mathbf{T}_N)}\text{adj}(^R\mathbf{T}_N) | ||
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\end{array}\right] | \end{array}\right] | ||
</math> | </math> | ||
+ | For a proof please have a look at the example in the main article of [[Matrix inversion|matrix inversion]]. | ||
}} | }} | ||
[[Category:Article]] | [[Category:Article]] | ||
[[Category:Matrices]] | [[Category:Matrices]] |
Latest revision as of 18:15, 13 November 2015
← Back: Gauß-Jordan-Algorithm | Overview: Matrix inversion | Next: Transformations → |
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. |