Cofactor Formula

Before going to learn the cofactor formula, let us see where it is used. The inverse of a matrix is used in solving a system of equations. The inverse of a matrix is its adjoint divided by its determinant. The adjoint of the matrix is the transpose of the cofactor matrix. The cofactor matrix is the matrix of the same order as the given matrix where the elements of the original matrix are replaced by their corresponding cofactors. These cofactors are calculated using the cofactor formula. Let us learn the cofactor formula along with a few solved examples.

What Is Cofactor Formula?

The cofactor \(C_{ij}\) of an element \(a_{ij}\) of a square matrix of order \(n \times n\) is its minor \(M_{ij}\) multiplied by \((-1)^{i+j}\). Here, the minor \(M_{ij}\) of the element \(a_{ij}\) is the determinant of the matrix obtained by removing the \(i\)^th row and \(j\)^th column from the original matrix. i.e.,

Cofactor of an element \(a_{ij}\) is, \(C_{ij} = (-1)^{i+j} \, \cdot M_{ij}\)

Here, \(M_{ij}\) = The minor of \(a_{ij}\) = The determinant of the matrix obtained by removing the \(i\)^th row and \(j\)^th column.

Note: 

i) The determinant of a 2 × 2 matrix \(\left[\begin{array}{ccc}a & b  \\ c & d \end{array}\right]\) is found using the formula, \(\left|\begin{array}{ccc}a & b  \\ c & d \end{array}\right|\) = \(ad - bc\).

ii) The cofactor of an element can be positive or negative or zero.

Examples Using Cofactor Formula

Example 1: Find the cofactor of 0 in the matrix \(\left[\begin{array}{ccc}1 & 0 & -2 \\ 3 & -1 & 2 \\ 4 & 5 & 6\end{array}\right]\).

Solution:

Before finding the cofactor of 0, we will first find its minor.

Minor of 0 = \(\left|\begin{array}{ccc}3 & 2  \\ 4 & 6 \end{array}\right|\) = 3(6) - 4(2) = 18 - 8 = 10.

0 is present in 1^st row and 2^nd column. So 

The cofactor of 0 = (-1)^{1 + 2} (10) = -10

Answer: The cofactor of 0 is -10.

Example 2: The adjoint of a matrix is the transpose of the cofactor matrix. Using this, find the adjoint of the matrix given in Example 1.

Solution:

The given matrix is \(\left[\begin{array}{ccc}1 & 0 & -2 \\ 3 & -1 & 2 \\ 4 & 5 & 6\end{array}\right]\).

We will replace each element of the given matrix with its cofactor to find the cofactor matrix.

The cofactor matrix = \(\left[\begin{array}{ccc}-16 & -10 & 19 \\ -10 & 14 & -5 \\ -2 & -8 & -1\end{array}\right]\).

The adjoint of a matrix is the transpose of the cofactor matrix. Thus,

The adjoint matrix of the given matrix =  \(\left[\begin{array}{ccc}-16 & -10 & -2 \\ -10 & 14 & -8 \\ 19 & -5 & -1\end{array}\right]\).

Answer: The adjoint of the given matrix is  \(\left[\begin{array}{ccc}-16 & -10 & -2 \\ -10 & 14 & -8 \\ 19 & -5 & -1\end{array}\right]\).