Difference between revisions of "Minors and cofactors"

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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]]}}
  
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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/>
  
 
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/>
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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><br/><br/>
  
 
{{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=
 +
<br/><math>
 +
\mathbf{A}_e  =
 +
\left[\begin{array}{ccc}
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1&0&1\\
 +
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}_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}
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\Box & \Box & \Box & \Box\\
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1&\Box&1\\
 +
\Box&\Box&\Box\\
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1&\Box&2
 +
\end{array}\right|=
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\left|\begin{array}{cc}
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1&1\\
 +
1&2
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\end{array}\right|=2-1=1
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</math><br/>
 +
<math>
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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
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</math><br/><br/>
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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/>
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<math>C_{31}(\mathbf{A}_e)=(-1)^{3+1}M_{31}(\mathbf{A}_e)=(-1)^4\cdot-1=-1</math>
 +
}}
 +
 
 +
{{Example
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|Title=Minors and cofactors of a 4-by-4 matrix
 
|Contents=
 
|Contents=
 
<br/><math>
 
<br/><math>
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0 & 0 & 2 & 1
 
0 & 0 & 2 & 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_{14}(\mathbf{A}_e)</math> and <math>M_{31}(\mathbf{A}_e)</math> for example are defined as<br/><br/>
−
<math>M_{1,4}(\mathbf{A}_e)=
+
<math>M_{14}(\mathbf{A}_e)=
 
\left|\begin{array}{cccc}
 
\left|\begin{array}{cccc}
 
\Box & \Box & \Box & \Box\\
 
\Box & \Box & \Box & \Box\\
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</math><br/>
 
</math><br/>
 
<math>
 
<math>
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M_{3,1}(\mathbf{A}_e)=
+
M_{31}(\mathbf{A}_e)=
 
\left|\begin{array}{cccc}
 
\left|\begin{array}{cccc}
 
\Box & 2 & 0 & 0\\
 
\Box & 2 & 0 & 0\\
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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_{1,4}(\mathbf{A}_e)=(-1)^{1+4}M_{1,4}(\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_{3,1}(\mathbf{A}_e)=(-1)^{3+1}M_{3,1}(\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 15:57, 22 May 2014

← 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_{i,j}(\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}_e = \left[\begin{array}{ccc} 1&0&1\\ 3&1&0\\ 1&0&2 \end{array}\right]

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

Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): 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
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

The corresponding cofactors in that case are

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

C_{31}(\mathbf{A}_e)=(-1)^{3+1}M_{31}(\mathbf{A}_e)=(-1)^4\cdot-1=-1

Example: Minors and cofactors of a 4-by-4 matrix


\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]

The minors M_{14}(\mathbf{A}_e) and M_{31}(\mathbf{A}_e) for example are defined as

M_{14}(\mathbf{A}_e)= \left|\begin{array}{cccc} \Box & \Box & \Box & \Box\\ 3 & 0 & 1 & \Box\\ 0 & 1 & 0 & \Box\\ 0 & 0 & 2 & \Box \end{array}\right|= \left|\begin{array}{ccc} 3 & 0 & 1\\ 0 & 1 & 0\\ 0 & 0 & 2 \end{array}\right|=6-0=6
M_{31}(\mathbf{A}_e)= \left|\begin{array}{cccc} \Box & 2 & 0 & 0\\ \Box & 0 & 1 & 1\\ \Box & \Box & \Box & \Box\\ \Box & 0 & 2 & 1 \end{array}\right|= \left|\begin{array}{ccc} 2 & 0 & 0\\ 0 & 1 & 1\\ 0 & 2 & 1 \end{array}\right|=2-4=-2

The corresponding cofactors in that case are

C_{14}(\mathbf{A}_e)=(-1)^{1+4}M_{14}(\mathbf{A}_e)=(-1)^5\cdot6=-6

C_{31}(\mathbf{A}_e)=(-1)^{3+1}M_{31}(\mathbf{A}_e)=(-1)^4\cdot-2=-2

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