Cofactor of a Matrix

A cofactor of a matrix is an important concept in linear algebra, and it is often used to calculate determinants and inverses of matrices. The cofactor of an element in a matrix is calculated as

1. Minor: For a given element a_ij​ in a matrix, first find the minor of that element. The minor of a_ij​, denoted as M_ij​, is the determinant of the matrix that remains after removing the i-th row and j-th column from the original matrix.
2. Cofactor: The cofactor of a_ij​, denoted as C_ij​, is then obtained by multiplying the minor M_ij​ by (−1)^i+j. This factor (−1)^i+j accounts for the sign change that depends on the position of the element.

The cofactors for the above matrix is given below:

• C₁₁= (-1)² (3 - 16) = -13
• C₂₁= (-1)³ (-1 - (-2)) = -1

For example minor of the element a₁₁ matrix \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}is calculated as:

M_{11} = det\begin{bmatrix} 5 & 6\\ 8 & 9 \end{bmatrix}\\ = 45-48\\ = -3

Formula

If we denote the Cofactor using C_ij, then the cofactor of any element for 

C_ij = M_ij × (-1)^i+j

Where,

• i is the number of rows for the element under consideration,
• j is the number of columns for the element under consideration, and
• M_ij is the minor of the element in the i^th row and j^th column.

How to Find

In order to find a cofactor matrix we need to perform the following steps:

• Step 1: Find the minor of each element of the matrix and make a minor matrix.
• Step 2: Multiply each element in the minor matrix by (-1)^i+j_.
  Thus, we obtain the cofactor matrix.

Let us understand how to find a cofactor matrix using an example:

Example: Find the cofactor matrix of \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}

Solution:

• Step 1: Find the minor of each element and make a minor matrix.

Minor of a₁₁ is calculated by eliminating row 1 and column 1 and taking the determinant of the remaining matrix as follows:

M₁₁= determinant of \begin{bmatrix} 5 & 6\\ 8 & 9 \end{bmatrix}
M₁₁ = 5(9) - 6(8)
M₁₁ = 45 - 48 = -3

Similarly, the minor of element a₁₂ is calculated:

M₁₂ = determinant of \begin{bmatrix} 4 & 6\\ 7 & 9 \end{bmatrix}
M₁₂ = 4(9) - 6(7)
M₁₂ = 36 - 42 = -6

• Similarly, calculate the minors of all elements to obtain the following minor matrix:

\begin{bmatrix} -3 & -6 & -3\\ -6 & -12 & -6\\ -3 & -6 & -3 \end{bmatrix}

• Step 2: Multiply each element of the minor matrix by (-1)^i+j to get the cofactor of that element i.e. C_ij

C₁₁ = M₁₁ × (-1)¹⁺¹ = M₁₁ = -3
C₁₂ = M₁₂ × (-1)¹⁺² = -M₁₂ = 6

• Similarly, calculate the other cofactors to obtain the following cofactor matrix:

\begin{bmatrix} -3 & 6 & -3\\ 6 & -12 & 6\\ -3 & 6 & -3 \end{bmatrix}

Applications

There are various applications of Cofactor Matrix. Some of these applications are:

• Cofactor of the Matrix is used to find the adjoint of the matrix.
• Cofactor Matrix is used in the calculation of determinant of the matrix.
• It is also used to find the inverse of the matrix

➢Practice: Solved Examples