Difference between revisions of "Minors and cofactors"

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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 smaller square matrix obtained by removing the row <math>i</math> and the column <math>j</math> from <math>\mathbf{A}</math>.<br/><br/>
 
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 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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<math>C_{i,j}(\mathbf{A})=(-1)^{i+j}M_{i,j}(\mathbf{A})</math><br/><br/>
  
 
{{Example
 
{{Example
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|Title=Minors
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|Title=Minors and cofactors
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|Contents=
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<br/><math>
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\mathbf{A}_e  =
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\left[\begin{array}{cccc}
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1 & 2 & 0 & 0\\
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3 & 0 & 1 & 1\\
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0 & 1 & 0 & 0\\
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0 & 0 & 2 & 1
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\end{array}\right]</math><br/><br/>
 
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/>
 
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/>
 
<math>M_{1,4}(\mathbf{A}_e)=
 
<math>M_{1,4}(\mathbf{A}_e)=
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\end{array}\right|=6-0=6
 
\end{array}\right|=6-0=6
 
</math><br/>
 
</math><br/>
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  <math>
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<math>
 
M_{3,1}(\mathbf{A}_e)=
 
M_{3,1}(\mathbf{A}_e)=
 
\left|\begin{array}{cccc}
 
\left|\begin{array}{cccc}
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0 & 2 & 1
 
0 & 2 & 1
 
\end{array}\right|=2-4=-2
 
\end{array}\right|=2-4=-2
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</math>
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</math><br/><br/>
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The corresponding cofactors in that case are<br/><br/>
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<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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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_{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_{i,j}(\mathbf{A})=(-1)^{i+j}M_{i,j}(\mathbf{A})</math><br/><br/>
+
}}
−
 
 
−
  <math>C_{1,4}(\mathbf{A}_e)=(-1)^{1+4}M_{1,4}(\mathbf{A}_e)=(-1)^5\cdot6=-6</math>
 

Revision as of 15:40, 9 May 2014

The minor M_{i,j}(\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_{i,j}(\mathbf{A})

Example: Minors and cofactors


\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_{1,4}(\mathbf{A}_e) and M_{3,1}(\mathbf{A}_e) for example are defined as

M_{1,4}(\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_{3,1}(\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_{1,4}(\mathbf{A}_e)=(-1)^{1+4}M_{1,4}(\mathbf{A}_e)=(-1)^5\cdot6=-6

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

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