How to Calculate the Cofactor: Step-by-Step Guide
A cofactor is a number derived by removing the row and column of a given element in the shape of a square or rectangle. Depending on the element's position, the cofactor is preceded by a negative or positive sign. It is used to find the inverse and adjoint of the matrix. In this article we will learn cofactor matrix, cofactor example and how to find cofactor of a matrix. We will also solve a few examples to understand the concept of cofactor of matrices .
How to Find the Cofactor of a Matrix ?
Let’s understand how to find a cofactor of a matrix
Consider the following matrix:
\[\begin{bmatrix} 6 & 4 & 3\\ 9 & 2 & 5\\ 1 & 7 & 8\end{bmatrix}\]
To find the cofactor of 2, we place blinders over the 2 and eliminate the rows and columns that include 2 as seen below:
\[\begin{bmatrix} 6 & 3\\ 1 & 8 \end{bmatrix}\]
Now we have a matrix that is missing two digits. The determinant of the matrix, which will be the cofactor of 2, will be easy to find. We get by multiplying the diagonal elements of the matrix.
6 x 8 = 48
3 x 1 = 3
Subtract the second diagonal value from the first., i.e, 48 - 3 = 45.
Examine the symbol that has been allocated to the phone number. Every 3 x 3 determinant has a sign based on the deleted element's position.
The cofactor matrix can be written using the matrix sign..
For 2x2 matrix sigen is given as
\[\begin{bmatrix} + & -\\ - & + \end{bmatrix}\]
Cofactor matrix 3x3 is given below as :
\[\begin{bmatrix} + & - & +\\ - & + & -\\ + & - & +\end{bmatrix}\]
Examine the exact location of the 2. It's worth noting that the positive symbol appears before the 2. As a result, the final value is +3, or 3.
Minors and Cofactors
A minor is the determinant of a square matrix that is produced by deleting a row and a column from a square matrix. The minors are determined by the deleted columns and rows. For example, if the fourth column and second row of the matrix are removed, the determinant of the matrix is M24.
So, in a matrix, which is basically a grid in the shape of a square or a rectangle, co-factors are the number you obtain when you remove the row and column of a designated element. Depending on whether the number is in the + or – position, the co-factor is always preceded by a negative (-) or positive (+) sign.
Cofactor Formula
Let A denote any n x n matrix, and Mij denote the (n-1) x (n- 1) matrix formed by removing the ith row and jth column. Then, det(M\[_{ij}\]) is called the minor of a\[_{ij}\]. The formula for calculating the cofactor C\[_{ij}\] of a\[_{ij}\] is:
C\[_{ij}\] = (-1)\[^{i+j}\] det(M\[_{ij}\])
As a result, the sign for cofactor is always +ve (positive) or -ve (negative).
Solved Examples:
1. Find the Cofactor Matrix of the Given Matrix
A = \[\begin{bmatrix} 1 & 9 & 3\\ 2 & 5 & 4\\ 3 & 7 & 8\end{bmatrix}\]
Sol: Given matrix is:
A = \[\begin{bmatrix} 1 & 9 & 3\\ 2 & 5 & 4\\ 3 & 7 & 8\end{bmatrix}\]
Let M\[_{ij}\] be the minor of the ith row and jth column items.
We have to find minor of the elements of matrix A. It is calculated below:
M\[_{11}\] = \[\begin{vmatrix} 5 & 4 \\ 7 & 8\end{vmatrix}\] = 40 - 28 = 12
M\[_{12}\] = \[\begin{vmatrix} 2 & 4 \\ 3 & 8\end{vmatrix}\] = 16 - 12 = 4
M\[_{13}\] = \[\begin{vmatrix} 2 & 5 \\ 3 & 7\end{vmatrix}\] = 14 - 15 = -1
M\[_{21}\] = \[\begin{vmatrix} 9 & 3 \\ 7 & 8\end{vmatrix}\] = 72 - 21 = 51
M\[_{22}\] = \[\begin{vmatrix} 1 & 3 \\ 3 & 8\end{vmatrix}\] = 8 - 9 = -1
M\[_{23}\] = \[\begin{vmatrix} 1 & 9 \\ 3 & 7\end{vmatrix}\] = 7 - 27 = -20
M\[_{31}\] = \[\begin{vmatrix} 9 & 3 \\ 5 & 4\end{vmatrix}\] = 36 - 15 = 21
M\[_{32}\] = \[\begin{vmatrix} 1 & 3 \\ 2 & 4\end{vmatrix}\] = 4 - 6 = -2
M\[_{33}\] = \[\begin{vmatrix} 1 & 9 \\ 2 & 5\end{vmatrix}\] = 5 - 18 = -13
Matrix of cofactors of A is: \[\begin{bmatrix} +12 & -4 & +(-1)\\ -51 & +(-1) & -(-20)\\ +21 & -(-2) & +(-13)\end{bmatrix}\]
= \[\begin{bmatrix} 12 & -4 & -1\\ -51 & -1 & 20\\ 21 & 2 & -13\end{bmatrix}\]
2. If the Cofactor of the Element a\[_{11}\] of the Matrix A = \[\begin{bmatrix} 2 & -3 & 5\\ 6 & 0 & p\\ 1 & 5 & -7\end{bmatrix}\] is -20, then Find the Value of p.
Sol: Given matrix is:
A = \[\begin{bmatrix} 2 & -3 & 5\\ 6 & 0 & p\\ 1 & 5 & -7\end{bmatrix}\]
Using the formula of cofactor of an element,
C\[_{ij}\] = (-1)\[^{i+j}\] det (M\[_{ij}\])
Cofactor of a\[_{11}\] is:
C\[_{11}\] = (-1)\[^{1+1}\] det (M\[_{11}\])
-20 = \[\begin{vmatrix} 0 & p \\ 5 & -7\end{vmatrix}\]
⇒ - 20 = 0 - 5p
⇒ 5p = 20
⇒ p = \[\frac{20}{5}\]
⇒ p = 4
Hence, the value of p is 4.
Conclusion:
From the above we have understood the concept of how to find cofactor. We have seen a cofactor method to calculate the cofactor of a matrix. We should note that If the elements of a row (or column) are multiplied with the cofactors of any other row (or column), then their sum is zero. There are cofactor matrix calculator available so that we can calculate the cofactor of the matrix.
FAQs on Cofactor: Meaning, Formula & Simple Examples
1. What is the definition of a cofactor for an element in a matrix?
In matrix algebra, the cofactor of an element is its corresponding minor multiplied by a specific sign. The minor is the determinant of the submatrix formed by removing the element's row and column. The sign is determined by the element's position (i, j) using the formula (-1)i+j. Therefore, a cofactor encapsulates both the value of the minor and its positional sign within the larger matrix.
2. What is the main difference between a minor and a cofactor in matrix theory?
The primary difference between a minor and a cofactor is the inclusion of a positional sign.
- A minor (Mij) of an element is simply the determinant of the smaller matrix that remains after deleting the i-th row and j-th column. It is always a numerical value without a specific sign attached.
- A cofactor (Cij) is the minor multiplied by (-1)i+j. This means the cofactor is a 'signed minor'. For some positions the cofactor is the same as the minor, while for others, it is the negative of the minor.
3. How do you calculate the cofactor of an element? Provide an example.
To calculate the cofactor of an element aij in a matrix, you follow two steps:
1. First, find the minor (Mij) by calculating the determinant of the submatrix created by removing row 'i' and column 'j'.
2. Then, apply the sign using the formula Cij = (-1)i+j * Mij.
Example: For element a11 in a 3x3 matrix, its cofactor C11 would be (-1)1+1 multiplied by the determinant of the 2x2 submatrix formed by removing the first row and first column.
4. What is a cofactor matrix, and how is it constructed from a given matrix?
A cofactor matrix is a square matrix of the same size as the original matrix, where each element at position (i, j) is replaced by its corresponding cofactor, Cij. To construct it, you must systematically calculate the cofactor for every single element of the original matrix and place it in the same position in the new matrix. This resulting matrix is crucial for finding the adjoint and inverse of the original matrix.
5. What is the significance of the sign convention, (-1)i+j, in the cofactor formula?
The sign convention (-1)i+j is fundamentally important because it creates a 'checkerboard' pattern of positive and negative signs across the matrix. This pattern is essential for the correct expansion of the determinant. Without these alternating signs, the calculation would merely sum up minors, leading to an incorrect determinant value and subsequently an incorrect matrix inverse. The sign ensures that each element's contribution to the determinant has the correct orientation and value as per the properties of determinants.
6. How are cofactors used to find the adjoint and inverse of a square matrix?
Cofactors are a critical step in finding the inverse of a matrix. The process is as follows:
- Step 1: Calculate the cofactor matrix by finding the cofactor for every element of the original matrix.
- Step 2: Find the adjoint of the matrix (often written as adj(A)), which is the transpose of the cofactor matrix.
- Step 3: The inverse of the matrix (A-1) is then calculated using the formula: A-1 = (1/det(A)) * adj(A). This shows that without cofactors, you cannot find the adjoint, and therefore cannot find the inverse of a matrix (for matrices 3x3 and larger).
7. What is the value of the determinant of a matrix in terms of its cofactors?
The determinant of a matrix can be defined as 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 (R1), the determinant of a 3x3 matrix A would be:
det(A) = a11C11 + a12C12 + a13C13
This property highlights the direct relationship between a matrix's elements, their cofactors, and the overall determinant value, and it forms the basis for the Laplace expansion method.
8. What happens if the elements of one row are multiplied by the cofactors of any other row?
A key property of cofactors, as per the CBSE Class 12 syllabus, states that if the elements of one row (or column) are multiplied by the cofactors of a different row (or column), the sum of these products will always be zero. For instance, if you multiply the elements of the first row (a11, a12, a13) with the cofactors of the second row (C21, C22, C23), the result (a11C21 + a12C22 + a13C23) will be 0. This property is crucial in understanding the structure and properties of determinants.
