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 formula for finding the cofactor of a matrix?
The formula of finding the cofactor of a matrix is Cij = (−1)i+j |Mij|
2. What is a matrix and a determinant?
A matrix is a rectangular or square array with a given number of rows and columns of either constants or variables.
A determinant is a scalar value that is obtained through the functions of elements of a square matrix.
3. What are the different operations that can be performed on the matrix?
The operations of addition, subtraction and multiplication can be performed between two or more matrices.
4. What is the minor of an element of a matrix?
A minor is the value that is obtained from the square matrix determinant by striking out rows and columns of the element whose minor is to be obtained.
5. What are the basic properties of the determinant of a matrix?
The basic properties of a determinant are
Det(I)=1, where I am the identity matrix of order n x n.
Det (AT)= Det(A) where AT is the transpose of the matrix A
Det(B-1) = 1/ det (B) where B-1 is the inverse of determinant B
Det (AB) = det(A) det(B) where A and B are two square matrices with the same order
Det (CY) = Cn det(Y) where C is the constant, n is the order of the matrix and Y is the matrix in question.
6. What is the difference between cofactors and minors of a matrix?
Minor of an element of a square matrix is the determinant that we get by deleting the row and the column in which the element appears. The cofactor of an element of a square matrix is the minor of the element with a proper sign. Suppose the element appears in the row and jth column. Then the appropriate sign of the element is . The sign thus obtained is to be multiplied with the minor of the element to get the corresponding cofactor.
7. How can i find if the cofactor of a matrix is correct?
An easy way could be to multiply the cofactors of any row/column to the elements of any other row/column except the row/column from which the cofactors were obtained. If this comes out to be zero, your cofactors were correct.
8. What is the practical use of matrices and determinants?
Matrices are used much more in daily life than people would have thought. It is in front of us every day when going to work, at the university and even at home. Graphic software such as Adobe Photoshop on your personal computer uses matrices to process linear transformations to render images. A square matrix can represent a linear transformation of a geometric object.