Difference between revisions of "Minors and cofactors"
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|Title=Minors and cofactors of a 4-by-4 matrix | |Title=Minors and cofactors of a 4-by-4 matrix | ||
|Contents= | |Contents= | ||
− | + | This example uses the transformation matrix <math>^R\mathbf{T}_N</math> that is introduced in the robotics script in chapter 3 on page 3-37 and used on the following pages. | |
− | <br/><math> | + | <br/> |
− | \mathbf{ | + | :<math> |
+ | ^R\mathbf{T}_N = | ||
\left[\begin{array}{cccc} | \left[\begin{array}{cccc} | ||
− | 1 | + | 0 & 1 & 0 & 2a\\ |
− | + | 0 & 0 & -1 & 0\\ | |
− | 0 | + | -1 & 0 & 0 & 0\\ |
− | 0 & 0 & | + | 0 & 0 & 0 & 1 |
\end{array}\right]</math><br/><br/> | \end{array}\right]</math><br/><br/> | ||
− | The minors <math>M_{ | + | The minors <math>M_{31}(^R\mathbf{T}_N)</math> and <math>M_{42}(^R\mathbf{T}_N)</math> for example are defined as<br/><br/> |
− | <math>M_{ | + | :<math> |
+ | M_{31}(^R\mathbf{T}_N)= | ||
\left|\begin{array}{cccc} | \left|\begin{array}{cccc} | ||
+ | \Box & 1 & 0 & 2a\\ | ||
+ | \Box & 0 & -1 & 0\\ | ||
\Box & \Box & \Box & \Box\\ | \Box & \Box & \Box & \Box\\ | ||
− | + | \Box & 0 & 0 & 1 | |
− | |||
− | 0 & | ||
\end{array}\right|= | \end{array}\right|= | ||
\left|\begin{array}{ccc} | \left|\begin{array}{ccc} | ||
− | + | 1&0&2a\\ | |
− | 0 & 1 & 0\\ | + | 0&-1&0\\ |
− | 0 & 0 & | + | 0&0&1 |
− | \end{array}\right|= | + | \end{array}\right|=-1-0=-1 |
− | </math | + | :</math> |
− | <math> | + | <math>M_{42}(^R\mathbf{T}_N)= |
− | M_{ | ||
\left|\begin{array}{cccc} | \left|\begin{array}{cccc} | ||
− | \Box & | + | 0&\Box&0&2a\\ |
− | \Box & 0 & | + | 0&\Box&-1&0\\ |
+ | -1&\Box&0&0 | ||
\Box & \Box & \Box & \Box\\ | \Box & \Box & \Box & \Box\\ | ||
− | |||
\end{array}\right|= | \end{array}\right|= | ||
\left|\begin{array}{ccc} | \left|\begin{array}{ccc} | ||
− | + | 0&0&2a\\ | |
− | 0 & 1 & | + | 0&-1&0\\ |
− | 0 & | + | -1&0&0 |
− | \end{array}\right|= | + | \end{array}\right|=0-2a=-2a |
− | </math><br/><br/> | + | </math><br/> |
+ | <br/><br/> | ||
The corresponding cofactors in that case are<br/><br/> | The corresponding cofactors in that case are<br/><br/> | ||
− | <math>C_{14}(\mathbf{A}_e)=(-1)^{1+4}M_{14}(\mathbf{A}_e)=(-1)^5\cdot6=-6</math><br/><br/> | + | :<math>C_{14}(\mathbf{A}_e)=(-1)^{1+4}M_{14}(\mathbf{A}_e)=(-1)^5\cdot6=-6</math><br/><br/> |
− | <math>C_{31}(\mathbf{A}_e)=(-1)^{3+1}M_{31}(\mathbf{A}_e)=(-1)^4\cdot-2=-2</math> | + | :<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 16:09, 22 May 2014
← Back: Determinant of a matrix | Overview: Matrices | Next: Matrix inversion → |
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
This example uses the transformation matrix that is introduced in the robotics script in chapter 3 on page 3-37 and used on the following pages.
The minors and for example are defined as Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): M_{42}(^R\mathbf{T}_N)= \left|\begin{array}{cccc} 0&\Box&0&2a\\ 0&\Box&-1&0\\ -1&\Box&0&0 \Box & \Box & \Box & \Box\\ \end{array}\right|= \left|\begin{array}{ccc} 0&0&2a\\ 0&-1&0\\ -1&0&0 \end{array}\right|=0-2a=-2a
|