Difference between revisions of "Minors and cofactors"
From Robotics
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|Title=Minors and cofactors of a 3-by-3 matrix | |Title=Minors and cofactors of a 3-by-3 matrix | ||
|Contents= | |Contents= | ||
− | <br/><math> | + | <br/> |
+ | :<math> | ||
\mathbf{A}_e = | \mathbf{A}_e = | ||
\left[\begin{array}{ccc} | \left[\begin{array}{ccc} | ||
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\end{array}\right]</math><br/><br/> | \end{array}\right]</math><br/><br/> | ||
The minors <math>M_{22}(\mathbf{A}_e)</math> and <math>M_{31}(\mathbf{A}_e)</math> for example are defined as<br/><br/> | The minors <math>M_{22}(\mathbf{A}_e)</math> and <math>M_{31}(\mathbf{A}_e)</math> for example are defined as<br/><br/> | ||
− | <math>M_{22}(\mathbf{A}_e)= | + | :<math>M_{22}(\mathbf{A}_e)= |
\left|\begin{array}{ccc} | \left|\begin{array}{ccc} | ||
− | |||
1&\Box&1\\ | 1&\Box&1\\ | ||
\Box&\Box&\Box\\ | \Box&\Box&\Box\\ | ||
Line 28: | Line 28: | ||
1&2 | 1&2 | ||
\end{array}\right|=2-1=1 | \end{array}\right|=2-1=1 | ||
− | </math><br/> | + | </math><br/><br/> |
− | <math> | + | :<math> |
M_{31}(\mathbf{A}_e)= | M_{31}(\mathbf{A}_e)= | ||
\left|\begin{array}{ccc} | \left|\begin{array}{ccc} | ||
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</math><br/><br/> | </math><br/><br/> | ||
The corresponding cofactors in that case are<br/><br/> | The corresponding cofactors in that case are<br/><br/> | ||
− | <math>C_{22}(\mathbf{A}_e)=(-1)^{2+2}M_{22}(\mathbf{A}_e)=(-1)^4\cdot1=1</math><br/><br/> | + | :<math>C_{22}(\mathbf{A}_e)=(-1)^{2+2}M_{22}(\mathbf{A}_e)=(-1)^4\cdot1=1</math><br/><br/> |
− | <math>C_{31}(\mathbf{A}_e)=(-1)^{3+1}M_{31}(\mathbf{A}_e)=(-1)^4\cdot-1=-1</math> | + | :<math>C_{31}(\mathbf{A}_e)=(-1)^{3+1}M_{31}(\mathbf{A}_e)=(-1)^4\cdot-1=-1</math> |
}} | }} | ||
Revision as of 15:59, 22 May 2014
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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 of a 3-by-3 matrix
The minors and for example are defined as The corresponding cofactors in that case are |
Example: Minors and cofactors of a 4-by-4 matrix
|