Cofactor

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.