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Minors and Cofactors

Matrices and determinants play a very important role in mathematics. A matrix is defined as a rectangular arrangement of expressions or numbers (called entries) in the form of horizontal and vertical arrays, referred as rows and columns. On the other hand, a determinant is said to be a value that is computed from a square matrix.
This value is calculated in the form of a specified arithmetic expression made from the entries of the square matrix. This technique of computing determinant is known as cofactor expansion.

The determinants give very useful information about the matrix of coefficients corresponding to a system of linear equations and about the matrix corresponding to linear transformation of a vector space.
We come across with several useful applications of the matrices and determinants. There are two important concepts related to matrices and determinants - minors and cofactors whose knowledge in compulsory in the computation of adjoint of matrix and hence in its inverse as well as in computation of determinant of square matrix.  In this page below, we are going to discuss what are these two concepts and how to find the values of minors and cofactors with the help of solved examples.

 

Definitions

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Minors
In a square matrix, each element possesses its own minor. A minor is defined as a value computed from the determinant of a square matrix which is obtained after crossing out a row and a column corresponding to the element that is under consideration. Let us consider that P be a square matrix. 

Suppose we want to compute the value of minor for $ij^{th}$ element (the element in $i^{th}$ row and $j^{th}$ column), then we should form a determinant of a sub-matrix obtained by hiding $i^{th}$ row and $j^{th}$ column. This minor is known as (i,,j) minor and is usually often denoted $M_{i,j}$.

In order to find the minors of some square matrix, we are supposed to block out a row and a column one by one at a time and calculate their determinant, until all the minors are computed.
We may follow the steps mentioned below to find each minor of given matrix:

Step 1: Hide each $i^{th}$ row and $j^{th}$ column one by one from given matrix, where i goes from 1 to m and j goes from 1 to n (m and n being total number of rows and columns in that matrix).

Step 2: Calculate the value of the determinant of the matrix made after hiding a row and
a column from step 1.

Cofactor
The cofactor is defined as the signed minor. An (i,,j) cofactor is computed by multiplying (i, j) minor by $(-1)^{i+j}$ and is denoted by $C_{ij}$.

In order to get the cofactors of a matrix, follow the steps mentioned below:

Step 1: Find each (i, j) minor where i and j vary from 1 to m and n respectively (m and n represent total number of rows and columns in given matrix).

Step 2: Now use the following formula to find the cofactors:

$C_{ij} = (-1)^{i + j}. Mij$

where, M_{ij} denotes the minor of $i^{th}$ row and $j^{th}$ column of the matrix.

It is quite evident that when value of (i + j) is even, the positive sign is applied to the minor and when the value of (i + j) is odd, the negative sign is applied.

2 X 2 Matrix

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Let us consider a 2 x 2 matrix as given below:

A = $\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}$

Minors:
When we hide a row and a column in such matrix, we end up getting a single number instead of a determinant or submatrix. i.e.

$M_{11} = \begin{bmatrix}
- & -\\
- & d
\end{bmatrix}$

= d

$M_{12} = \begin{bmatrix}
- & -\\
c & -
\end{bmatrix}$

= c

$M_{21} = \begin{bmatrix}
- & b\\
- & -
\end{bmatrix}$

= b

$M_{22} = \begin{bmatrix}
a & -\\
- & -
\end{bmatrix}$

= a

Cofactors:

The cofactors are obtained as shown below:

$C_{11} = (-1)^{1+1}M_{11}$

$C_{11} = (-1)^{2} d$ = d

$C_{12} = (-1)^{1+2}M_{12}$

$C_{12} = (-1)^{3} c$ = -c

$C_{21} = (-1)^{2+1}M_{21}$

$C_{12} = (-1)^{3} b$ = -b

$C_{22} = (-1)^{2+2}M_{22}$

$C_{22} = (-1)^{4} a$ = a

3 X 3 Matrix

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Let us assume a 3 x 3 matrix as shown below:

A = $\begin{bmatrix}
a & b & c\\
d & e & f\\
g & h & i
\end{bmatrix}$

Minors:

$M_{11} = \begin{vmatrix}
e & f\\
h & i
\end{vmatrix}$

= ei - fh

$M_{12} = \begin{vmatrix}
d & f\\
g & i
\end{vmatrix}$

= di - fg

$M_{13} = \begin{vmatrix}
d & e\\
g & h
\end{vmatrix}$

= dh - eg

$M_{21} = \begin{vmatrix}
b & c\\
h & i
\end{vmatrix}$

= bi - ch

$M_{22} = \begin{vmatrix}
a & c\\
g & i
\end{vmatrix}$

= ai - cg

$M_{23} = \begin{vmatrix}
a & b\\
g & h
\end{vmatrix}$

= ah - bg

$M_{31} = \begin{vmatrix}
b & c\\
e & f
\end{vmatrix}$

= bf - ce

$M_{32} = \begin{vmatrix}
a & c\\
d & f
\end{vmatrix}$

= af - cd

$M_{33} = \begin{vmatrix}
a & b\\
d & e
\end{vmatrix}$

= ae - bd

Cofactors:

$C_{11} = (-1)^{1+1}M_{11}$

$C_{11} = (-1)^{2} (ei - fh)$ = ei - fh

$C_{12} = (-1)^{1+2}M_{12}$

$C_{12} = (-1)^{3} (di - fg)$ = fg - di

$C_{13} = (-1)^{1+3}M_{13}$

$C_{13} = (-1)^{4} (dh - eg)$ = dh - eg

$C_{21} = (-1)^{2+1}M_{21}$

$C_{21} = (-1)^{3} (bi - ch)$ = ch - bi

$C_{22} = (-1)^{2+2}M_{22}$

$C_{22} = (-1)^{4} (ai - cg)$ = ai - cg

$C_{23} = (-1)^{2+3}M_{23}$

$C_{23} = (-1)^{5} (ah - bg)$ = bg - ah

$C_{31} = (-1)^{3+1}M_{31}$

$C_{31} = (-1)^{4} (bf - ce)$ = bf - ce

$C_{32} = (-1)^{3+2}M_{32}$

$C_{32} = (-1)^{5} (af - cd)$ = cd - af

$C_{33} = (-1)^{3+3}M_{33}$

$C_{33} = (-1)^{6} (ae - bd)$ = ae - bd

4 X 4 Matrix

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Let us assume a 4 x 4 matrix as written below:

$A = \begin{bmatrix}
a & b & c & d\\
e & f & g & h\\
i & j & k & l\\
m & n & o & p
\end{bmatrix}$

Minors:

$M_{11} = \begin{vmatrix}
f & g & h\\
j & k & l\\
n & o & p
\end{vmatrix}$

= f(kp - lo) - g(jp - ln) + h(jo - kn)

Similarly, other minors can be calculated.

Cofactors:

$C_{11} = (-1)^{1+1} M_{11}$

= f(kp - lo) - g(jp - ln) + h(jo - kn)

Other cofactors can also be calculated in the similar manner.

Examples

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Have a look at the following examples for finding minor and cofactors:

Example 1: Find the minors and cofactors of :

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

Minors:

$M_{11}$ = 1
$M_{12}$ = 0
$M_{21}$ = -4
$M_{22}$ = 5

Cofactors:

$C_{11} = (-1)^{1+1}M_{11} = (-1)^{2} 1$ = 1

$C_{12} = (-1)^{1+2}M_{12} = (-1)^{3} 0$ = 0

$C_{21} = (-1)^{2+1}M_{21} = (-1)^{3} (-4)$ = 4

$C_{22} = (-1)^{2+2}M_{22} = (-1)^{4} 5$ = 5

Example 2: Find the minor matrix of the following matrix :

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

Solution : $M_{11} = \begin{vmatrix}
-1 & 1\\
1 & -1
\end{vmatrix}$

= 1 - 1 = 0

$M_{12} = \begin{vmatrix}
2 & 1\\
1 & -1
\end{vmatrix}$

= -2 - 1 = -3

$M_{13} = \begin{vmatrix}
2 & -1\\
1 & 1
\end{vmatrix}$

= 2 + 1 = 3

$M_{21} = \begin{vmatrix}
-1 & 2\\
1 & -1
\end{vmatrix}$

= 1 - 2 = -1

$M_{22} = \begin{vmatrix}
1 & 2\\
1 & -1
\end{vmatrix}$

= -1 - 2 = -3

$M_{23} = \begin{vmatrix}
1 & -1\\
1 & 1
\end{vmatrix}$

= 1 + 1 = 2

$M_{31} = \begin{vmatrix}
-1 & 2\\
-1 & 1
\end{vmatrix}$

= -1 + 2 = 1

$M_{32} = \begin{vmatrix}
1 & 2\\
2 & 1
\end{vmatrix}$

= 1 - 4 = -3

$M_{33} = \begin{vmatrix}
1 & -1\\
2 & -1
\end{vmatrix}$

= -1 + 2 = 1

Thus, minor matrix is given below:

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