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

Minors and Cofactors Matrix

Now let’s come to what is 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 matrix 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 = \[\begin{vmatrix} 24 & 5 & -4)\\ -12& 3 & 2\\ -2 & -5 & 4 \end{vmatrix}\]


Example 2. Find the cofactor matrix of A given that

A = \[\begin{vmatrix} 1 & 2 & 3\\ 4& 5 &6 \\ 7& 8 & 9 \end{vmatrix}\]

Solution 2) Let \[M_{ij}\] be the minor of every element

\[M_{11}\] = \[\begin{vmatrix} 5 & 6\\ 8 & 9 \end{vmatrix}\] = 45 - 48 = -3

\[M_{12}\] = \[\begin{vmatrix} 4 & 6\\ 7 & 9 \end{vmatrix}\] = 36 - 42 = -6

\[M_{13}\] = \[\begin{vmatrix} 4 & 5\\ 7 & 8 \end{vmatrix}\] = 32 - 35 = -3

\[M_{21}\] = \[\begin{vmatrix} 2 & 3\\ 8 & 9 \end{vmatrix}\] = 18 - 24 = -6

\[M_{22}\] = \[\begin{vmatrix} 1 & 3\\ 7 & 9 \end{vmatrix}\] = 9 - 27 = -12

\[M_{23}\] = \[\begin{vmatrix} 1 & 2\\ 7 & 8 \end{vmatrix}\] = 8 -14 = -16

\[M_{31}\] = \[\begin{vmatrix} 2 & 3\\ 5 & 6 \end{vmatrix}\] = 12 - 15 = -3

\[M_{32}\] = \[\begin{vmatrix} 1 & 3\\ 4 & 6 \end{vmatrix}\] = 6 - 12 = -6

\[M_{33}\] = \[\begin{vmatrix} 1 & 2\\ 4 & 5 \end{vmatrix}\] = 5 - 8 = -3

The cofactor matrix A is  

A = \[\begin{vmatrix} +(-3) & -(-6) & +(-3))\\ -(-6)& +(-12) & -(-6)\\ +(-3) & -(6) & +(-3) \end{vmatrix}\]

A = \[\begin{vmatrix} -3 & 6 & -3\\ 6& -12 & 6\\ -3& 6 & -3 \end{vmatrix}\]

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

FAQ (Frequently Asked Questions)

1. 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.

2.  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.

3. 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.