Difference between revisions of "Minors and cofactors"
From Robotics
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The '''minor <math>M_{i,j}(\mathbf{A})</math>''' of an n-by-n square matrix <math>\mathbf{A}</math> is the determinant of a smaller square matrix obtained by removing the row <math>i</math> and the column <math>j</math> from <math>\mathbf{A}</math>.<br/><br/> | The '''minor <math>M_{i,j}(\mathbf{A})</math>''' of an n-by-n square matrix <math>\mathbf{A}</math> is the determinant of a smaller square matrix obtained by removing the row <math>i</math> and the column <math>j</math> from <math>\mathbf{A}</math>.<br/><br/> | ||
+ | |||
+ | Multiplying the minor with <math>(-1)^{i+j}</math> results in the '''cofactor <math>C_{i,j}(\mathbf{A})</math>''':<br/><br/> | ||
+ | <math>C_{i,j}(\mathbf{A})=(-1)^{i+j}M_{i,j}(\mathbf{A})</math><br/><br/> | ||
{{Example | {{Example | ||
− | |Title=Minors | + | |Title=Minors and cofactors |
− | + | |Contents= | |
+ | <br/><math> | ||
+ | \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]</math><br/><br/> | ||
The minors <math>M_{1,4}(\mathbf{A}_e)</math> and <math>M_{3,1}(\mathbf{A}_e)</math> for example are defined as<br/><br/> | The minors <math>M_{1,4}(\mathbf{A}_e)</math> and <math>M_{3,1}(\mathbf{A}_e)</math> for example are defined as<br/><br/> | ||
<math>M_{1,4}(\mathbf{A}_e)= | <math>M_{1,4}(\mathbf{A}_e)= | ||
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\end{array}\right|=6-0=6 | \end{array}\right|=6-0=6 | ||
</math><br/> | </math><br/> | ||
− | + | <math> | |
M_{3,1}(\mathbf{A}_e)= | M_{3,1}(\mathbf{A}_e)= | ||
\left|\begin{array}{cccc} | \left|\begin{array}{cccc} | ||
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0 & 2 & 1 | 0 & 2 & 1 | ||
\end{array}\right|=2-4=-2 | \end{array}\right|=2-4=-2 | ||
− | </math> | + | </math><br/><br/> |
− | + | The corresponding cofactors in that case are<br/><br/> | |
− | <br/> | + | <math>C_{1,4}(\mathbf{A}_e)=(-1)^{1+4}M_{1,4}(\mathbf{A}_e)=(-1)^5\cdot6=-6</math><br/><br/> |
− | + | <math>C_{3,1}(\mathbf{A}_e)=(-1)^{3+1}M_{3,1}(\mathbf{A}_e)=(-1)^4\cdot-2=-2</math> | |
− | <math>C_{ | + | }} |
− | |||
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Revision as of 15:40, 9 May 2014
The minor of an n-by-n square matrix is the determinant of a smaller square matrix obtained by removing the row and the column from .
Multiplying the minor with results in the cofactor :
Example: Minors and cofactors
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