Difference between revisions of "Minors and cofactors"
From Robotics
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{{Navigation|before=[[Determinant of a matrix]]|overview=[[Matrices]]|next=[[Matrix inversion]]}} | {{Navigation|before=[[Determinant of a matrix]]|overview=[[Matrices]]|next=[[Matrix inversion]]}} | ||
− | The '''minor <math>M_{ | + | The '''minor <math>M_{ij}(\mathbf{A})</math>''' of an n-by-n square matrix <math>\mathbf{A}</math> is the [[Determinant of a matrix|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/> | 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_{ | + | <math>C_{i,j}(\mathbf{A})=(-1)^{i+j}M_{ij}(\mathbf{A})</math><br/><br/> |
{{Example | {{Example | ||
− | |Title=Minors and cofactors | + | |Title=Minors and cofactors of a 3-by-3 matrix |
+ | |Contents= | ||
+ | <br/><math> | ||
+ | \mathbf{A}_e = | ||
+ | \left[\begin{array}{ccc} | ||
+ | 1&0&1\\ | ||
+ | 3&1&0\\ | ||
+ | 1&0&2 | ||
+ | \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/> | ||
+ | <math>M_{22}(\mathbf{A}_e)= | ||
+ | \left|\begin{array}{ccc} | ||
+ | \Box & \Box & \Box & \Box\\ | ||
+ | 1&\Box&1\\ | ||
+ | \Box&\Box&\Box\\ | ||
+ | 1&\Box&2 | ||
+ | \end{array}\right|= | ||
+ | \left|\begin{array}{cc} | ||
+ | 1&1\\ | ||
+ | 1&2 | ||
+ | \end{array}\right|=2-1=1 | ||
+ | </math><br/> | ||
+ | <math> | ||
+ | M_{31}(\mathbf{A}_e)= | ||
+ | \left|\begin{array}{ccc} | ||
+ | \Box & 0&1\\ | ||
+ | \Box & 1&0\\ | ||
+ | \Box & \Box & \Box\\ | ||
+ | \end{array}\right|= | ||
+ | \left|\begin{array}{cc} | ||
+ | 0&1\\ | ||
+ | 1&0 | ||
+ | \end{array}\right|=0-1=-1 | ||
+ | </math><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_{31}(\mathbf{A}_e)=(-1)^{3+1}M_{31}(\mathbf{A}_e)=(-1)^4\cdot-1=-1</math> | ||
+ | }} | ||
+ | |||
+ | {{Example | ||
+ | |Title=Minors and cofactors of a 4-by-4 matrix | ||
|Contents= | |Contents= | ||
<br/><math> | <br/><math> | ||
Line 17: | Line 57: | ||
0 & 0 & 2 & 1 | 0 & 0 & 2 & 1 | ||
\end{array}\right]</math><br/><br/> | \end{array}\right]</math><br/><br/> | ||
− | The minors <math>M_{ | + | The minors <math>M_{14}(\mathbf{A}_e)</math> and <math>M_{31}(\mathbf{A}_e)</math> for example are defined as<br/><br/> |
− | <math>M_{ | + | <math>M_{14}(\mathbf{A}_e)= |
\left|\begin{array}{cccc} | \left|\begin{array}{cccc} | ||
\Box & \Box & \Box & \Box\\ | \Box & \Box & \Box & \Box\\ | ||
Line 32: | Line 72: | ||
</math><br/> | </math><br/> | ||
<math> | <math> | ||
− | M_{ | + | M_{31}(\mathbf{A}_e)= |
\left|\begin{array}{cccc} | \left|\begin{array}{cccc} | ||
\Box & 2 & 0 & 0\\ | \Box & 2 & 0 & 0\\ | ||
Line 46: | Line 86: | ||
</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_{ | + | <math>C_{14}(\mathbf{A}_e)=(-1)^{1+4}M_{14}(\mathbf{A}_e)=(-1)^5\cdot6=-6</math><br/><br/> |
− | <math>C_{ | + | <math>C_{31}(\mathbf{A}_e)=(-1)^{3+1}M_{31}(\mathbf{A}_e)=(-1)^4\cdot-2=-2</math> |
}} | }} | ||
[[Category:Article]] | [[Category:Article]] | ||
[[Category:Matrices]] | [[Category:Matrices]] |
Revision as of 15:57, 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
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Example: Minors and cofactors of a 4-by-4 matrix
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