Minors and Cofactors are two of the most important concepts in matrices. They are essential for finding the adjoint and the inverse of a matrix. To find the determinants of a large square matrix (such as 4x4), it is important to calculate the minors and then the cofactors of that matrix. Here is a thorough explanation of “what are minors and cofactors” along with the steps to find them.

Topics in Determinants

Introduction to Determinants

Minors and Cofactors

Properties of Determinants

System of Linear Equations using Determinants

Differentiation and Integration of Determinants

Standard Determinants

Minors are individuals under the age of 18 who are not considered adults under the law.

The minor of an element in a matrix, e.g. ${a}{22}$ in the determinant $\begin{vmatrix} {a}{11} & {a}{12} & {a}{13} \ {a}{21} & {a}{22} & {a}{23} \ {a}{31} & {a}{32} & {a}{33} \ \end{vmatrix}$, is defined as the determinant obtained by deleting the row and column in which that element lies.

The minor of A12 is denoted as M12.

Here, $$M_{12} = \left| \begin{matrix} a_{21} & a_{23} \ a_{31} & a_{33} \ \end{matrix} \right|$$

Cofactors are molecules that are required for an enzyme to work properly. They can be either inorganic ions or organic molecules. Cofactors can either be bound to the enzyme’s active site, or they can be free in the cell. In either case, they are essential for the enzyme to catalyze a reaction.

The relation between the cofactor of an element (a_{ij}) and its minor is given by:

({C}{i,j}={{\left( -1 \right)}^{i+j}}{{M}{i,j}}),

where (i) denotes the (i^{th}) row and (j) denotes the (j^{th}) column to which the element (a_{ij}) belongs.

The value of the determinant of order three can be defined in terms of ‘Minor’ and ‘Cofactor’ as:

(D = a_{11}M_{11} - a_{12}M_{12} + a_{13}M_{13} ;;;;; \text{or} ;;;;; D = a_{11}C_{11} - a_{12}C_{12} + a_{13}C_{13})

Note: (a) A determinant of order 3 will have 9 minors, each of which will be a determinant of order 2, and a determinant of order 4 will have 16 minors, each of which will be a determinant of order 3.

(b) (\begin{array}{l}{{a}_{11}}{{C}_{21}}+{{a}_{12}}{{C}_{22}}+{{a}_{13}}{{C}_{23}}=0\end{array}), i.e. multiplying the cofactor of each row/column element results in a zero value.

Row and Column Operations on Determinants

This notation is used when we interchange the $i^{th}$ row (or column) and the $j^{th}$ row (or column), when $i \neq j$: $R_i \leftrightarrow R_j$ or $C_i \leftrightarrow C_j$.

This converts the corresponding column into the row: (\begin{array}{l}{{R}_{i}}\leftrightarrow {{C}_{i}};\end{array} )

This represents multiplying the ith row (or column) by k, represented as (R_{i} \to R_{k_{i}}) or (C_{i} \to kC_{i}), with (k \in R).

(d) (\begin{array}{l}{{R}_{i}}\to {{R}_{i}}k+{{R}_{j}};;;;or;;;Ci\to {{C}_{i}}k+{{C}_{j}};\left( i\ne j \right);\end{array} ) This symbol is used to multiply the $i^{th}$ row (or column) by $k$ and add the $j^{th}$ row (or column) to it.

Related Articles:

Matrices

Types of Matrices

Adjoint and Inverse of a Matrix

Finding Minors and Cofactors: Practice Problems

Question 1: The cofactor of a12 in the given matrix is -60.

Given:

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Solution:

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Eliminate all the elements of the first row and the second column, then calculate the determinant of the remaining elements to find the cofactor of $a_{12}$.

Here a12 = -3, element of the first row and second column.

M12 = Minor of a12 $$=\begin{vmatrix} 6 & 4\ 1& -7 \end{vmatrix} = 6(-7) - 4(1)$$

-42 - 4 = -46

Cofactor of $(-3) = (-1) \cdot 1 + 2 \cdot (-46) = -(-46) = 46$

Answer 2:

The minors and cofactors of the elements of the following determinants are as follows:

(\begin{array}{l}(i) \left| \begin{matrix} 2 & -4 \ 0 & 3 \ \end{matrix} \right| ;;;;; = 6 \\ (ii) \left| \begin{matrix} a & c \ b & d \ \end{matrix} \right| = ad-bc \end{array} )

Given:

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Solution:

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The remaining after eliminating a row and column of an element is the minor of the element.

$\left| \begin{matrix} 2 & -4 \ 0 & 3 \ \end{matrix} \right| = M_{11} = 3$ (Minor of element (2) = 3)

(\begin{array}{l}\text{Cofactor of}\ 2= (-1)^{2}M_{11}=3\\end{array} )

$M_{12} = Minor ; of ; element ; (-4) = \left| \begin{matrix} 2 & \cdots & -4 \ \vdots & & \vdots \ 0 & \cdots & 3 \end{matrix} \right| = 0; \ Cofactor ; of ; (-4) = (-1)^{1+2}M_{12} = (-1)0 = 0$

$$\begin{array}{l}{M_{21}}= Minor;;; of;; element \left( 0 \right)=\left| \begin{matrix} 2,,,-4 \ \vdots ,,,,,,,,, \ 0…,,3 \ \end{matrix} \right|=-4;;;; \Cofactor;; of ;;\left( 0 \right)={{\left( -1 \right)}^{2+1}}{{M}_{21}}=\left( -1 \right)\left( -4 \right)=4\\end{array} $$

$$M_{22} = Minor; of; element \left( 3 \right)=\left| \begin{matrix} 2,,,,-4 \ ,,,,,,,,,,\vdots \ 0…,,,3 \ \end{matrix} \right|=2; ;\Cofactor ;;of ;\left( 3 \right)={{\left( -1 \right)}^{2+2}}{{M}_{22}}=+2$$

(ii) (\begin{array}{l}\left| \begin{matrix} a & c \ b & d \ \end{matrix} \right|;,{M}_{11}= Minor; of; element ;\left( a \right)=d;\; Cofactor; of ;\left( a \right)={{\left( -1 \right)}^{1+1}}{{M}_{11}}={{\left( -1 \right)}^{2}}d=d\\end{array} )

$$\begin{array}{l}{M_{12}}=; Minor ;of; element ;(c) =\left| \begin{matrix} a\dots c \ ,,,,,,\vdots \ b,,,,d \ \end{matrix} \right|=b;; Cofactor ;of; (c) ={{\left( -1 \right)}^{1+2}}{{M}_{12}}={{\left( -1 \right)}^{3}}b=-b\\end{array} $$

$$\begin{array}{l}{M_{21}} = \text{Minor of element }\left( b \right) = \left| \begin{matrix} a,,,,,c \ \vdots ,,,,,,, \ b,…,d \ \end{matrix} \right| = c; ; \text{Cofactor of } \left( b \right) = {{\left( -1 \right)}^{2+1}}{{M}_{21}} = {{\left( -1 \right)}^{3}}c = -c\\end{array}$$

$\begin{array}{l}{M_{22}}=; Minor; of; element ;\left( d \right)=\left| \begin{matrix} a,,,,,c \ ,,,,,,,\vdots \ b…,,d \ \end{matrix} \right|=a; ;Cofactor ;of ;\left( d \right)={{\left( -1 \right)}^{2+2}}{{M}_{22}}={{\left( -1 \right)}^{4}}a=a\end{array}$

Question 3: Find the minor and cofactor of each element of the determinant (\begin{vmatrix} 2 & -2 & 3 \ 1 & 4 & 5 \ 2 & 1 & -3 \ \end{vmatrix} )

Given:

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Solution:

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The minor of an element can be found by eliminating the row and column of that element from the determinant. The cofactor of the element can be found by using the formula (\begin{array}{l}{{\left( -1 \right)}^{i+j}}{{M}_{i\times j}}\end{array} ).

The minors are:

• ${M}_{11}=\left| \begin{matrix} 4 & 5 \ 1 & -3 \ \end{matrix} \right|=-17$
• ${M}_{12}=\left| \begin{matrix} 1 & 5 \ 2 & -3 \ \end{matrix} \right|=-13$
• ${M}_{13}=\left| \begin{matrix} 1 & 4 \ 2 & 1 \ \end{matrix} \right|=-7$

(\begin{array}{l}{{M}_{21}}=\left| \begin{matrix} -2 & 3 \ 1 & -3 \ \end{matrix} \right|=3,;;; {{M}_{22}}=\left| \begin{matrix} 2 & 3 \ 2 & -3 \ \end{matrix} \right|=-12,;;; {{M}_{23}}=\left| \begin{matrix} 2 & -2 \ 2 & 1 \ \end{matrix} \right|=6\end{array})

$$M_{31}=\left| \begin{matrix} -2 & 3 \ 4 & 5 \ \end{matrix} \right|=-22, ;;;M_{32}=\left| \begin{matrix} 2 & 3 \ 1 & 5 \ \end{matrix} \right|=7, ;;;M_{33}=\left| \begin{matrix} 2 & -2 \ 1 & 4 \ \end{matrix} \right|=10$$

The cofactors are:

({A}{11}=-M{11}=-17,\{A}{12}=-M{12}=13,\{A}{13}=-M{13}=-7)

(\begin{array}{l}{{A}_{21}}=-{{M}_{21}}=-3,\{{A}_{22}}=-{{M}_{22}}=-12,\{{A}_{23}}=-{{M}_{23}}=-6\end{array} )

(\begin{array}{l}{{A}_{31}}=-{{M}_{31}}=-22,\{{A}_{32}}=-{{M}_{32}}=-7,\{{A}_{33}}={{M}_{33}}=10\end{array} )

Frequently Asked Questions

No, cofactor and minor are not the same.

The cofactor of an element $a_{ij}$ is defined by $C_{ij} = (-1)^{i+j} M_{ij}$, where $M_{ij}$ is the minor of $a_{ij}$, which is the determinant obtained by deleting the row and column in which that element lies.

A 3x3 matrix has 9 minors.

A 3x3 matrix has 9 minors.

How do you calculate the minor of an element in a matrix?

The minor of an element is calculated by deleting the row and column in which that element lies, and then finding the determinant of the resulting matrix.

The formula to find the cofactor of an element in a matrix is: C_ij = (-1)^i+j * det(A_ij)

The cofactor of an element $a_{ij}$ is given by $C_{ij} = (-1)^{i+j}M_{ij}$, where $M_{ij}$ is the minor of $a_{ij}$.

Applications of Minors and Cofactors of a Matrix

• Determining the Inverse of a Matrix
• Calculating the Determinant of a Matrix
• Solving Systems of Linear Equations
• Finding Eigenvalues and Eigenvectors

We use minors and cofactors to find the adjoint and inverse of matrices.

Finding the Cofactor Matrix

1. Begin by writing out the original matrix.
2. Find the determinant of the matrix.
3. Create a matrix of the cofactors.
4. Take the transpose of the cofactor matrix.

We need to calculate the minors of all the elements of the matrix first. This is done by deleting the row and column that the elements belong to, and then finding the determinant by considering the remaining elements. After that, we can find the cofactor of the elements. This is done by multiplying the minor of the element with -1i+j. If Mij is the minor, then the cofactor, Cij = -1i+jMij. Finally, we can form the cofactor matrix with the obtained values.

To find the adjoint matrix of a 2x2 matrix, use the formula:

$$adj(A) = \begin{bmatrix} \text{det}(A_{22}) & -\text{det}(A_{12}) \ -\text{det}(A_{21}) & \text{det}(A_{11}) \end{bmatrix}$$

We interchange the elements on the main diagonal (a11 and a22) and then put a negative sign for the elements at a12 and a21 positions. The resulting matrix is the adjoint of the given 2×2 matrix.

How many minors are in a 2x2 matrix?

A 2×2 matrix has 4 minors.