Formula for Cofactor of a Matrix
If we denote the Cofactor using Cij, then the cofactor of any element for
Cij = Mij × (-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
- Mij is the minor of the element in the ith row and jth column.
What is Minor?
A minor of a particular element is obtained by eliminating the row and column of the matrix to which that particular element belongs and then finding the determinant of the remaining part. The matrix formed by combining all the minors is called the minor matrix. For example minor of the element a11 matrix [Tex]\begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix} [/Tex]is calculated as:
[Tex]M_{11} = det\begin{bmatrix} 5 & 6\\ 8 & 9 \end{bmatrix}\\ = 45-48\\ = -3 [/Tex]
Cofactor of a Matrix: Formula and Examples
The cofactor of a Matrix is an important concept in linear algebra. It is often used in calculating the determinant and the inverse of a matrix. Cofactor of Matrix or Cofactor matrix is the matrix formed by the Cofactor of each element of any matrix where cofactor is a number that is obtained by multiplying the minor of the element of any given matrix with -1 raised to the power of the sum of the row and column number to which that element belongs. This matrix formed with the cofactor of each element is called the Cofactor Matrix or Cofactor of Matrix.
Cofactor of a Matrix
Cofactors and Minors are very important concepts in Linear Algebra which further help us find the determinants, adjoint, and inverse of the matrix as well. We have explained in detail about Cofactor of a Matrix.