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Minors and Cofactors

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A "minor" is the determinant of the square matrix formed by deleting one row and one column from some larger square matrix. Since there are lots of rows and columns in the original matrix, you can make lots of minors from it.
 
These minors are labelled according to the row and column you deleted. Useful for computing both the determinant and inverse of square matrices.


The cofactor of the element a$_{ij}$ is its minor prefixing:
We use +ve sign if  i+j  is even and −ve sign if  i+j  is odd .

Definition

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To find the minors of any matrix, expand block out every row and column one at a time until all the minors are found.  To find each minor of a matrix

1) Delete the ith row and jth column of the matrix.
2) Compute the determinant of the remaining matrix after deleting the row and column of step 1.
Cofactors:

To find the cofactors of a matrix, just use the minors and apply the following formula:
C$_{ij}$ = (-1)$^{i + j}$M$_{ij}$

where M$_{ij}$ is the minor in the ith row, jth position of the matrix:

Let A =  $\begin{bmatrix}
a_{11} &a_{12}  &a_{13} \\
a_{21} & a_{22} & a_{23}\\
a_{31} &a_{32}  & a_{33}
\end{bmatrix}$

be a 3*3 matrix then

cofactor matrix or cof(A) =  $\begin{bmatrix}
A_{11} &A_{12}  &A_{13} \\
A_{21} & A_{22} & A_{23}\\
A_{31} &A_{32}  & A_{33}
\end{bmatrix}$

Co factor of  3 * 3 Matrix

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Cofactor of any matrix can be found with the help of minor. To find the minors of any matrix, expand block out every row and column until all the minors are found.

Below are some steps to find the cofactor of matrix:

Step 1: First find the cofactors for the given matrix.
Cofactor C$_{ij}$ of A

Step 2: For an n by n matrix A, we define the cofactor of a matrix by

C$_{ij}$ = (-1)$^{i + j}$ M$_{ij}$

Step 3: This is a minor with a positive or negative sign attached.
If i + j is even,it's positive
if i + j is odd, then it is said to be negative.

Let us see with the help of an example how to find cofactors of any matrix.

Example 1:  Find the cofactor for the matrix below.

$\begin{bmatrix}
1 & -1 & 2 \\
2 & -1 & 1\\
1 & 1 & -1
\end{bmatrix}$

Solution:

Let A = $\begin{bmatrix}
1 & -1 & 2 \\
2 & -1 & 1\\
1 & 1 & -1
\end{bmatrix}$

Firstly find the minors of A
M$_{11}$ = Minor of 1 = $\begin{vmatrix}
-1  & 1\\
 1& -1
\end{vmatrix}$

M$_{11}$= (-1)(-1) - 1(1)
= 0

M$_{12}$ = Minor of - 1 = $\begin{vmatrix}
2 & 1\\
1 & -1
\end{vmatrix}$
= -3

M$_{13}$ = Minor of 2 = $\begin{vmatrix}
2 & - 1\\
 1& 1
\end{vmatrix}$
= 3

M$_{21}$ = Minor of 2 = $\begin{vmatrix}
 -1& 2\\
 1& -1
\end{vmatrix}$
= - 1

M$_{22}$  = Minor of - 1 = $\begin{vmatrix}
1 & 2\\
1 & -1
\end{vmatrix}$
= -3

M$_{23}$ = Minor of 1 = $\begin{vmatrix}
 1& -1\\
 1& 1
\end{vmatrix}$
=2

M$_{31}$ = Minor of 1 = $\begin{vmatrix}
 - 1& 2\\
-1  & 1
\end{vmatrix}$
= 1

M$_{32}$ = Minor of 1 = $\begin{vmatrix}
1 & 2\\
 2& 1
\end{vmatrix}$
 = -3

M$_{33}$ = Minor of -1 = $\begin{vmatrix}
1 & -1\\
2 & -1
\end{vmatrix}$
= 1
Now it becomes very much easier for us to find their cofactors.

Cofactor of 1   = (-1)$^{1 + 1}$ M$_{11}$
= 0

Cofactor of -1   = (-1)$^{1 + 2}$ M$_{12}$
= 3

Cofactor of 2   = (-1)$^{1 + 3}$ M$_{13}$
= 3

Cofactor of 2   = (-1)$^{2 + 1}$ M$_{21}$
= 1

Cofactor of -1   = (-1)$^{2 + 2}$ M$_{22}$
= - 3

Cofactor of 1   = (-1)$^{2 + 3}$ M$_{23}$
= -2

Cofactor of 1   = (-1)$^{3 + 1}$ M$_{31}$
= 1

Cofactor of 1   = (-1)$^{3 + 2}$ M$_{32}$
= 3

Cofactor of -1   = (-1)$^{3 + 3}$ M$_{33}$
= 1

Cofactors of matrix = $\begin{bmatrix}
0 & 3 & 3 \\
1 & -3 & -2\\
1 & 3 & 1
\end{bmatrix}$

Cofactor of 4 * 4 matrix

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Cofactor of any matrix can be found with the help of minor. To find the minors of any matrix, expand block out every row and column until all the minors are found. Let us see with the help of an example how to find cofactors of any matrix.

Let us find the cofactors for the matrix:

Example 1: Find the cofactor of the below 4 * 4 matrix.

$\begin{bmatrix}
 1& 4 & -1  & 0\\
 2& 3 & 5 & -2 \\
 0  &  3& 1 & 6 \\
 3& 0 & 2 & 1
\end{bmatrix}$

Solution:  Cofactor of each element of A

A$_{11}$ = (-1)$^{1 + 1}$ $\begin{vmatrix}
3 & 5 & -2 \\
3 & 1 & 6\\
0 & 2 & 1
\end{vmatrix}$
= - 60

A$_{12}$ = (-1)$^{1 + 2}$ $\begin{vmatrix}
2 & 5 & -2 \\
0 & 1 & 6\\
3 & 2 & 1
\end{vmatrix}$
= - 74

A$_{13}$ = (-1)$^{1 + 3}$ $\begin{vmatrix}
2 & 3 & -2 \\
0 & 3 & 6\\
3 & 0 & 1
\end{vmatrix}$
= 78

A$_{14}$ = (-1)$^{1 + 4}$ $\begin{vmatrix}
2 & 3 & 5 \\
0 & 3 & 1\\
3 & 0 & 2
\end{vmatrix}$
= 24

A$_{21}$ = (-1)$^{2 + 1}$ $\begin{vmatrix}
4 & -1 & 0 \\
3 & 1 & 6\\
0 & 2 & 1
\end{vmatrix}$
=41

A$_{22}$ = (-1)$^{2+2}$ $\begin{vmatrix}
1 & -1 & 0 \\
0 & 1 & 6\\
3& 2 & 1
\end{vmatrix}$
= - 29

A$_{23}$ = (-1)$^{2+3}$ $\begin{vmatrix}
1 & 4 & 0 \\
0 & 3 & 6\\
3 & 0 & 1
\end{vmatrix}$
= - 75

A$_{24}$ = (-1)$^{2+4}$ $\begin{vmatrix}
1 & 4 & -1 \\
0 & 3 & 1\\
3 & 0 & 2
\end{vmatrix}$
 = 27

A$_{31}$ = (-1)$^{3 + 1}$ $\begin{vmatrix}
4 & -1 & 0 \\
3 & 5 & -2\\
0 & 2 & 1
\end{vmatrix}$
= 39

A$_{32}$ = (-1)$^{3+2}$ $\begin{vmatrix}
1 & -1 & 0 \\
2 & 5 & -2\\
3 & 2 & 1
\end{vmatrix}$
= -17

A$_{33}$ = (-1)$^{3+3}$ $\begin{vmatrix}
1 & 4 & 0 \\
2 & 3 & -2\\
3 & 0 & 1
\end{vmatrix}$
= -29

A$_{34}$ = (-1)$^{3+4}$ $\begin{vmatrix}
1 & 4 & -1 \\
2 & 3 & 5\\
3 & 0 & 2
\end{vmatrix}$
= - 59

A$_{41}$ = (-1)$^{4 + 1}$ $\begin{vmatrix}
4 & -1 & 0 \\
3 & 5 & -2\\
0 & 2 & 1
\end{vmatrix}$
= - 152

A$_{42}$ = (-1)$^{4 + 2}$ $\begin{vmatrix}
1 & -1 & 0 \\
2 & 5 & -2\\
3 & 2 & 1
\end{vmatrix}$
= 44

A$_{43}$ = (-1)$^{4+3}$ $\begin{vmatrix}
1 & 4 & 0 \\
2 & 3 & -2\\
3 & 0 & 1
\end{vmatrix}$
= 24

A$_{44}$ = (-1)$^{4+4}$ $\begin{vmatrix}
1 & 4 & -1 \\
2 & 3 & 5\\
3 & 0 & 2
\end{vmatrix}$
= - 26

Therefore the cofactor matrix of A = $\begin{bmatrix}
-60 & -74 & 78 & 24\\
41 & -29 & -75 & 27\\
39 & -17 &-29  &-59 \\
 -152&  44& 24 & -26
\end{bmatrix}$

Examples

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Given below are some examples on minors and cofactors.

Example 1: Find the cofactor of the below matrix.

$\begin{bmatrix}
 1&  2\\
 8 & 9
\end{bmatrix}$

Solution: Let A = $\begin{bmatrix}
 1&  2\\
 8 & 9
\end{bmatrix}$

Step 1: Find minor of A

M$_{11}$ = (Minor of 1) = 9

M$_{12}$ = (Minor of 2) = 8

M$_{21}$ = (Minor of 8) = 2

M$_{22}$ = (Minor of 9) = 1

Step 2: Co factor of 1 = (-1)1 + 1 M$_{11}$ = 9

Co factor of 2 = (-1)1 + 2 M$_{12}$ = -8

Co factor of 8 = (-1)2 + 1 M$_{21}$ =  -2

Co factor of 9 = (-1)2 + 2 M$_{11}$ = 1

The cofactor matrix A = $\begin{bmatrix}
 9&  -8\\
 -2 & 1
\end{bmatrix}$