Minors and Cofactors of Determinants
- Definition: The minor of an element in a matrix is the determinant of the smaller matrix obtained by removing the row and column containing that element.
For a matrix with elements , the minor of , denoted as , is the determinant of the matrix formed by deleting the th row and th column.
- Definition: Cofactors are scalar values associated with each element of a matrix, obtained by multiplying the minor of the element by , where and are the row and column indices, respectively.
For a matrix with elements , the cofactor of , denoted as , is given by , where is the minor of .
Properties and Significance:
Determinant Calculation: Cofactors play a crucial role in calculating the determinant of a matrix using the formula (expanding along any row or column).
Adjoint and Inverse Matrices: Cofactors are used in the calculation of the adjoint matrix and subsequently in finding the inverse of a matrix.
Matrix Properties: Minors and cofactors provide insights into the properties of matrices, especially regarding invertibility and determinant properties.
Consider a matrix :
The minor (the minor of the element in the second row, first column) is obtained by removing the second row and first column and calculating its determinant.
The cofactor of the element is obtained by multiplying the minor by because the row index + column index is odd).
Determinants: Minors and cofactors are essential in expanding determinants by minors or cofactors.
Matrix Properties: They contribute to understanding matrix properties, including invertibility, adjoint calculation, and system solvability.