Difference between revisions of "Minors and cofactors"

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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/>
 
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/>
 
<math>C_{i,j}(\mathbf{A})=(-1)^{i+j}M_{ij}(\mathbf{A})</math><br/><br/>
 
<math>C_{i,j}(\mathbf{A})=(-1)^{i+j}M_{ij}(\mathbf{A})</math><br/><br/>
  

Revision as of 16:13, 22 May 2014

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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}_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

M_{22}(\mathbf{A}_e)= \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}_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

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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