Difference between revisions of "Minors and cofactors"

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{{Navigation|before=[[Multiplication of matrices]]|overview=[[Matrices]]|next=[[Determinant of a matrix]]}}
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{{Navigation|before=[[Determinant of a matrix]]|overview=[[Matrices]]|next=[[Matrix inversion]]}}
  
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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 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/>
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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/>
  
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Multiplying the minor with <math>(-1)^{i+j}</math> results in the '''cofactor <math>C_{i,j}(\mathbf{A})</math>''':<br/><br/>
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Multiplying the minor with <math>(-1)^{i+j}</math> results in the '''cofactor <math>C_{ij}(\mathbf{A})</math>''':<br/><br/>
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<math>C_{i,j}(\mathbf{A})=(-1)^{i+j}M_{i,j}(\mathbf{A})</math><br/><br/>
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<math>C_{i,j}(\mathbf{A})=(-1)^{i+j}M_{ij}(\mathbf{A})</math>
  
 
{{Example
 
{{Example
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|Title=Minors and cofactors
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|Title=Minors and cofactors of a 3-by-3 matrix
 
|Contents=
 
|Contents=
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<br/><math>
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<br/>
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\mathbf{A}_e  =  
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:<math>
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\mathbf{A}_3  =
 +
\left[\begin{array}{ccc}
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1&0&1\\
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3&1&0\\
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1&0&2
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\end{array}\right]</math><br/><br/>
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The minors <math>M_{22}(\mathbf{A}_3)</math> and <math>M_{31}(\mathbf{A}_3)</math> for example are defined as<br/><br/>
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:<math>M_{22}(\mathbf{A}_3)=
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\left|\begin{array}{ccc}
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1&\Box&1\\
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\Box&\Box&\Box\\
 +
1&\Box&2
 +
\end{array}\right|=
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\left|\begin{array}{cc}
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1&1\\
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1&2
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\end{array}\right|=2-1=1
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</math><br/><br/>
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:<math>
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M_{31}(\mathbf{A}_3)=
 +
\left|\begin{array}{ccc}
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\Box & 0&1\\
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\Box & 1&0\\
 +
\Box & \Box & \Box\\
 +
\end{array}\right|=
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\left|\begin{array}{cc}
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0&1\\
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1&0
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\end{array}\right|=0-1=-1
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</math><br/><br/>
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The corresponding cofactors in that case are<br/><br/>
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:<math>C_{22}(\mathbf{A}_3)=(-1)^{2+2}M_{22}(\mathbf{A}_e)=(-1)^4\cdot1=1</math><br/><br/>
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:<math>C_{31}(\mathbf{A}_3)=(-1)^{3+1}M_{31}(\mathbf{A}_e)=(-1)^4\cdot-1=-1</math>
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}}
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{{Example
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|Title=Minors and cofactors of a 4-by-4 matrix
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|Contents=
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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.
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<br/>
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:<math>
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^R\mathbf{T}_N =  
 
\left[\begin{array}{cccc}
 
\left[\begin{array}{cccc}
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1 & 2 & 0 & 0\\
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0 & 1 & 0 & 2a\\
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3 & 0 & 1 & 1\\
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0 & 0 & -1 & 0\\
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0 & 1 & 0 & 0\\
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-1 & 0 & 0 & 0\\
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0 & 0 & 2 & 1
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0 & 0 & 0 & 1
 
\end{array}\right]</math><br/><br/>
 
\end{array}\right]</math><br/><br/>
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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/>
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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/>
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<math>M_{1,4}(\mathbf{A}_e)=
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:<math>
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M_{31}(^R\mathbf{T}_N)=
 
\left|\begin{array}{cccc}
 
\left|\begin{array}{cccc}
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\Box & 1 & 0 & 2a\\
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\Box & 0 & -1 & 0\\
 
\Box & \Box & \Box & \Box\\
 
\Box & \Box & \Box & \Box\\
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3 & 0 & 1 & \Box\\
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\Box & 0 & 0 & 1
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0 & 1 & 0 & \Box\\
 
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0 & 0 & 2 & \Box
 
 
\end{array}\right|=
 
\end{array}\right|=
 
\left|\begin{array}{ccc}
 
\left|\begin{array}{ccc}
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3 & 0 & 1\\
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1&0&2a\\
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0 & 1 & 0\\
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0&-1&0\\
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0 & 0 & 2
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0&0&1
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\end{array}\right|=6-0=6
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\end{array}\right|=-1-0=-1
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</math><br/>
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</math><br/><br/>
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<math>
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:<math>
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M_{3,1}(\mathbf{A}_e)=
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M_{42}(^R\mathbf{T}_N)=
 
\left|\begin{array}{cccc}
 
\left|\begin{array}{cccc}
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\Box & 2 & 0 & 0\\
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0&\Box&0&2a\\
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\Box & 0 & 1 & 1\\
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0&\Box&-1&0\\
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-1&\Box&0&0\\
 
\Box & \Box & \Box & \Box\\
 
\Box & \Box & \Box & \Box\\
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\Box & 0 & 2 & 1
 
 
\end{array}\right|=
 
\end{array}\right|=
 
\left|\begin{array}{ccc}
 
\left|\begin{array}{ccc}
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2 & 0 & 0\\
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0&0&2a\\
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0 & 1 & 1\\
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0&-1&0\\
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0 & 2 & 1
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-1&0&0
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\end{array}\right|=2-4=-2
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\end{array}\right|=0-2a=-2a
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</math><br/><br/>
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</math><br/>
 +
<br/><br/>
 
The corresponding cofactors in that case are<br/><br/>
 
The corresponding cofactors in that case are<br/><br/>
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<math>C_{1,4}(\mathbf{A}_e)=(-1)^{1+4}M_{1,4}(\mathbf{A}_e)=(-1)^5\cdot6=-6</math><br/><br/>
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:<math>C_{31}(^R\mathbf{T}_N)=(-1)^{3+1}M_{31}(^R\mathbf{T}_N)=(-1)^4\cdot(-1)=-1</math>
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<math>C_{3,1}(\mathbf{A}_e)=(-1)^{3+1}M_{3,1}(\mathbf{A}_e)=(-1)^4\cdot-2=-2</math>
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:<math>C_{42}(^R\mathbf{T}_N)=(-1)^{4+2}M_{42}(^R\mathbf{T}_N)=(-1)^6\cdot(-2a)=-2a</math>
 
}}
 
}}
  
 
[[Category:Article]]
 
[[Category:Article]]
 
[[Category:Matrices]]
 
[[Category:Matrices]]

Latest revision as of 18:10, 13 November 2015

← Back: Determinant of a matrix Overview: Matrices Next: Matrix inversion →

The minor M_{ij}(\mathbf{A}) of an n-by-n square matrix \mathbf{A} is the determinant of a smaller square matrix obtained by removing the row i and the column j from \mathbf{A}.

Multiplying the minor with (-1)^{i+j} results in the cofactor C_{ij}(\mathbf{A}):

C_{i,j}(\mathbf{A})=(-1)^{i+j}M_{ij}(\mathbf{A})

Example: Minors and cofactors of a 3-by-3 matrix


\mathbf{A}_3 = \left[\begin{array}{ccc} 1&0&1\\ 3&1&0\\ 1&0&2 \end{array}\right]

The minors M_{22}(\mathbf{A}_3) and M_{31}(\mathbf{A}_3) for example are defined as

M_{22}(\mathbf{A}_3)= \left|\begin{array}{ccc} 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

M_{31}(\mathbf{A}_3)= \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

The corresponding cofactors in that case are

C_{22}(\mathbf{A}_3)=(-1)^{2+2}M_{22}(\mathbf{A}_e)=(-1)^4\cdot1=1

C_{31}(\mathbf{A}_3)=(-1)^{3+1}M_{31}(\mathbf{A}_e)=(-1)^4\cdot-1=-1
Example: Minors and cofactors of a 4-by-4 matrix

This example uses the transformation matrix ^R\mathbf{T}_N that is introduced in the robotics script in chapter 3 on page 3-37 and used on the following pages.

^R\mathbf{T}_N = \left[\begin{array}{cccc} 0 & 1 & 0 & 2a\\ 0 & 0 & -1 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 \end{array}\right]

The minors M_{31}(^R\mathbf{T}_N) and M_{42}(^R\mathbf{T}_N) for example are defined as

M_{31}(^R\mathbf{T}_N)= \left|\begin{array}{cccc} \Box & 1 & 0 & 2a\\ \Box & 0 & -1 & 0\\ \Box & \Box & \Box & \Box\\ \Box & 0 & 0 & 1 \end{array}\right|= \left|\begin{array}{ccc} 1&0&2a\\ 0&-1&0\\ 0&0&1 \end{array}\right|=-1-0=-1

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



The corresponding cofactors in that case are

C_{31}(^R\mathbf{T}_N)=(-1)^{3+1}M_{31}(^R\mathbf{T}_N)=(-1)^4\cdot(-1)=-1
C_{42}(^R\mathbf{T}_N)=(-1)^{4+2}M_{42}(^R\mathbf{T}_N)=(-1)^6\cdot(-2a)=-2a
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