Introduction to Cofactor of Matrices
In many economic analyses, we assume the variables to relate to sets of linear equations. Matrix provides a clear and concise way to solve complex problems, many of which would be complicated using old algebraic methods. When we talk about matrices and determinants, minors and cofactors matrix is the most crucial concept relating to matrices. So, the main question is, what is cofactor? We use the cofactor matrix to find relevant information such as the adjoint and inverse of a matrix. To solve determinants, we use the concept of minors and cofactors to solve the problem. Before we start learning about minors and cofactors, let us brush up on determinants and matrices.
Matrix
It is a set of mxn numbers, whether the numbers are real or complex, arranged in a rectangular format, and having m rows and n columns and enclosed by a brackets is called mxn matrix.
An mxn matrix is expressed as
A = \[\begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21}& a_{22} &a_{23} \\ a_{31} &a_{32} & a_{33} \end{bmatrix}\]
The letters stand for real numbers. Note that is the element whose value represents ith row and jth column of the matrix. Thus, the matrix A is sometimes denoted by simplified form as \[(a_{ij})\] or by \[{a_{ij}}\], i.e., A = ( \[a_{ij}\] ). We usually denote matrices by the capital letters A, B, C, etc. We denote the elements as small letters a, b, c, etc.
Determinants
The determinant of a matrix is a scalar (number) obtained from the matrix element by specified operations, which is characteristic of the matrix. The determinants can be used only for square matrices. We denote it by det A or |A| for a square matrix A.
The determinant of the (2 x 2) matrix
A = \[\begin{bmatrix} a_{11} & a_{12}\\ a_{21}& a_{22} \end{bmatrix}\]
Is given by det A
|A| = \[\begin{vmatrix} a_{11} & a_{12}\\ a_{21}& a_{22} \end{vmatrix}\]
= a11* a22- a12* a21
What are the Different Types of Matrices?
Mentioned below is a brief understanding of the different types of matrices:
Row matrix- This type of matrix has only one row (1 x n)
Column matrix- This type of matrix has only one column (m x 1)
Null/Zero matrix- In this type of matrix all the elements of the matrix are zero i.e., aij=0
Horizontal matrix- It is a matrix in which the number of columns is more than the number of rows i.e., n>m
Vertical matrix- It is a matrix in which the number of rows is more than the number of columns i.e., m>n
Square matrix- In this type of matrix there are an equal number of rows and columns i.e., n=m
What are the Operations that are Performed in a Matrix?
The three main operations that are performed on matrices are addition, subtraction and multiplication.
The Operation of Addition of Matrices
If A and B are two matrices having the same number of rows and columns, or as we can call the same order, then A+B is equal to the formation of a third matrix with the same order as corresponding values of the two matrices are added.
The operation of the addition of matrices follows the commutative, associative, identity and additive inverse law.
The Operation of Subtraction of Matrices
Similarly, if A and B are two matrices with the same order then A-B will give a third matrix with the same order as corresponding values of the two matrices are subtracted. The value obtained can either be positive or negative.
The Operation of Multiplication of Matrices
Suppose there are two matrices A and B of order m x n and n x p respectively. To determine the product of these two matrices will be the value obtained of m x p.
The elements of every row of the matrix A or the first matrix need to be multiplied with the elements of every column of the matrix B or the second matrix. Then the products obtained are added together and placed respectively starting from the first element of the first row and the first column.
The operation of multiplication of matrices follows the commutative, associative, distributive, multiplicative identity, multiplicative inverse, cancellation and null matrix laws.
What is the Inverse of the Matrix?
Usually, the inverse of a matrix is obtained only if the given matrix is a square matrix with equal number of rows and columns. If the given matrix is A, for example, then its inverse would be A-1. Only when a matrix is applied the property of AA-1 = A-1A = I, it is proved that A-1 is the inverse of the matrix A.
What is the Transpose of a Matrix?
The matrix that is obtained by interchanging or reversing the rows with the column is called the transpose of a matrix. It is indicated by AT.
What is the Scalar Multiplication of a Matrix?
The scalar value, also known as the non-zero constant, is usually denoted by k. When each element of the matrix is multiplied by this value k it is called the scalar multiplication of the given matrix A by a nonzero constant k.
Minors and Cofactors Matrix
Now let’s come to what cofactor and minors. The cofactor definition is straightforward. A cofactor is a number that you will get when you remove the column and row of a value in a matrix. It is essential to properly understand minors and cofactor matrices so that you can solve complex problems relating to determinants. Now that we have understood the cofactor definition and meaning, you will be able to answer the question, what is cofactor? Now, we will see how to find the cofactor of a matrix. Here is a detailed method on how to find the cofactor of a matrix.
How to Find Cofactor?
In a given determinant, the minorMijof the element aijis the determinant of order (n – 1 x n – 1), which is obtained when we delete the ith row and jth column of Anxn.
For example, in the determinant
|A| = \[\begin{vmatrix} a_{11} & a_{12} & a_{13}\\ a_{21}& a_{22} &a_{23} \\ a_{31} &a_{32} & a_{33} \end{vmatrix}\]
The minor of the element a11 is
M11 = \[\begin{vmatrix} a_{22} & a_{23}\\ a_{32}& a_{33} \end{vmatrix}\]
The minor of the element a12 is
M12 = \[\begin{vmatrix} a_{21} & a_{23}\\ a_{32}& a_{33} \end{vmatrix}\]
The minor of the element a13 is
M13 = \[\begin{vmatrix} a_{21} & a_{22}\\ a_{31}& a_{32} \end{vmatrix}\]
The scalars \[C_{ij} = (-1)^{i+j}M_{ij}\] are called the cofactor of the element aijof the matrix A. Note: The value of the determinant can also be found by its minor elements or cofactors, as
\[a_{11}M_{11}\] - \[a_{12}M_{12}\] + \[a_{13}M_{13}\] or \[a_{11}C_{11}\] - \[a_{12}C_{12}\] + \[a_{13}C_{13}\]
Hence, det A is the sum of the elements of any row or column multiplied by their corresponding cofactors. We can find the value of the determinant if we expand it from any row or column.
\[\begin{vmatrix} + & -\\ - & + \end{vmatrix}\]
Scalars for 2x2 matrices.
\[\begin{vmatrix} + & - & +\\ - & + & -\\ + & - & + \end{vmatrix}\]
Scalars for 3x3 matrices.
(Image will be added soon)
The image depicts the scalars for MxM matrices.
We have now seen how to find the cofactor of a matrix. Now that you know how to use the cofactor method to solve problems, we will go through some cofactor examples.
Solved Examples
Example 1. Find the cofactor matrix of A given that
A = \[\begin{bmatrix} 1 &2 & 3\\ 0& 4 &5 \\ 1& 0 & 6 \end{bmatrix}\]
Solution 1) Let \[M_{ij}\] be the minor of every element
\[M_{11}\] = \[\begin{vmatrix} 4 & 5\\ 0 & 6 \end{vmatrix}\] = 24 - 0 = 24
\[M_{12}\] = \[\begin{vmatrix} 0 & 5\\ 1 & 6 \end{vmatrix}\] = 0 - 5 = -5
\[M_{13}\] = \[\begin{vmatrix} 0 & 4\\ 1 & 0 \end{vmatrix}\] = 0 - 4 = - 4
\[M_{21}\] = \[\begin{vmatrix} 2 & 3\\ 0 & 6 \end{vmatrix}\] = 12 - 0 = 12
\[M_{22}\] = \[\begin{vmatrix} 1 & 3\\ 1 & 6 \end{vmatrix}\] = 6 - 3 = 3
\[M_{23}\] = \[\begin{vmatrix} 1 & 2\\ 1 & 0 \end{vmatrix}\] = 0 - 2 = -2
\[M_{31}\] = \[\begin{vmatrix} 2 & 3\\ 4 & 5 \end{vmatrix}\] = 10 - 12 = -2
\[M_{32}\] = \[\begin{vmatrix} 1 & 3\\ 0 & 5 \end{vmatrix}\] = 5 - 0 = 5
\[M_{33}\] = \[\begin{vmatrix} 1 & 2\\ 0 & 4 \end{vmatrix}\] = 4 - 0 = 4
The cofactor matrix A is
A = \[\begin{vmatrix} +(24) & -(-5) & +(-4))\\ -(12)& +(3) & -(-2)\\ +(-2) & -(5) & +(4) \end{vmatrix}\]
A = \[\begin{vmatrix} 24 & 5 & -4)\\ -12& 3 & 2\\ -2 & -5 & 4 \end{vmatrix}\]
Matrices and Determinants Application
Matrices and determinants are widely used as they can help solve complex problems which include complex equations. Due to this, we use them in almost every field of science. Matrices give very compact ways of putting together a lot of information. They become vital for many applications in physics and engineering when you have formulas that depend on multi-dimensional quantities. Previously we would write it as an enormous number of separate equations, but nowadays, it can often be written down as just one matrix equation. Some of the areas where we can use matrices and determinants are as follows:
Statistics
Linear Programming
Optimization
Genetics
Robotics
Intersections of planes
FAQs on Cofactor of Matrices
1. What is the definition of a cofactor for an element in a matrix?
A cofactor of an element aij in a square matrix is a signed version of its minor. It is calculated by multiplying the minor of the element, Mij, by (-1) raised to the power of the sum of its row (i) and column (j) indices. The formula is Cij = (-1)i+j Mij. Cofactors are fundamental for calculating the determinant and the inverse of a matrix.
2. How is the cofactor of an element different from its minor?
The key difference lies in the sign.
- The minor (Mij) of an element is the value of the determinant of the sub-matrix formed by deleting the i-th row and j-th column.
- The cofactor (Cij) is the minor multiplied by a positional sign, determined by (-1)i+j. This results in a checkerboard pattern of positive and negative signs.
3. What is the step-by-step process to find the cofactor of an element in a 3x3 matrix?
To find the cofactor of any element aij in a 3x3 matrix, follow these steps:
- Identify the Element: Locate the element aij whose cofactor you need to find (e.g., a23, the element in the 2nd row, 3rd column).
- Calculate the Minor (Mij): Temporarily delete the entire row 'i' and column 'j' of the element. The determinant of the remaining 2x2 matrix is the minor, Mij.
- Determine the Sign: Calculate the sign using the formula (-1)i+j. For a23, the sign is (-1)2+3 = (-1)5 = -1.
- Calculate the Cofactor (Cij): Multiply the minor by the sign: Cij = (-1)i+j Mij.
4. How do you find the cofactors for a simple 2x2 matrix?
For a 2x2 matrix A =
$$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$
The cofactors are much simpler to find because the minor of each element is just a single element from the matrix. The cofactors are:
- C11 (for element a) = (-1)1+1 * d = d
- C12 (for element b) = (-1)1+2 * c = -c
- C21 (for element c) = (-1)2+1 * b = -b
- C22 (for element d) = (-1)2+2 * a = a
5. Why is the sign component (-1)i+j necessary in the cofactor formula?
The sign component (-1)i+j is crucial because it ensures the correct calculation of the matrix's determinant using the cofactor expansion method (also known as Laplace expansion). The determinant is not just a sum of products of elements and their minors; it's an alternating sum. This checkerboard pattern of signs (+, -, +, -, etc.) properly accounts for the geometric and algebraic properties of the determinant, ensuring the final scalar value is correct.
6. How are cofactors used to find the determinant of a matrix?
Cofactors provide a systematic way to calculate the determinant of any square matrix. This method, called cofactor expansion, states that the determinant is the sum of the products of the elements of any single row or column with their corresponding cofactors. For example, expanding along the first row of a 3x3 matrix A:
det(A) = a11C11 + a12C12 + a13C13.
Choosing a row or column with the most zeros simplifies this calculation significantly.
7. What is the relationship between the cofactor matrix, the adjoint, and the inverse of a matrix?
These three concepts are directly linked in a sequence that is vital for solving systems of linear equations:
- Cofactor Matrix: First, you find the cofactor for every element of the original matrix A and assemble them into a new matrix, called the cofactor matrix.
- Adjoint of a Matrix: The adjoint of A, denoted as adj(A), is the transpose of the cofactor matrix.
- Inverse of a Matrix: Finally, the inverse of A, denoted as A-1, is calculated by dividing the adjoint of A by the determinant of A. The formula is: A-1 = (1/det(A)) * adj(A). This shows that calculating cofactors is a critical step in finding a matrix's inverse.
8. Is it possible to calculate cofactors for a non-square matrix?
No, it is not possible to calculate cofactors for a non-square (rectangular) matrix. The concept of a cofactor is intrinsically linked to the determinant of a matrix. Since determinants are only defined for square matrices (where the number of rows equals the number of columns), the building blocks for cofactors—minors—cannot be calculated for non-square matrices. Therefore, cofactors are exclusively a property of square matrices.
