Minors and Cofactors of Determinants

Minors:

  • 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 A with elements aij, the minor of aij, denoted as Mij, is the determinant of the matrix formed by deleting the ith row and jth column.

Cofactors:

  • Definition: Cofactors are scalar values associated with each element of a matrix, obtained by multiplying the minor of the element by (1)i+j, where i and j are the row and column indices, respectively.

For a matrix A with elements aij, the cofactor of aij, denoted as Cij, is given by Cij=(1)i+j×Mij, where Mij is the minor of aij.

Properties and Significance:

  1. Determinant Calculation: Cofactors play a crucial role in calculating the determinant of a matrix using the formula det(A)=j=1naij×Cij (expanding along any row or column).

  2. Adjoint and Inverse Matrices: Cofactors are used in the calculation of the adjoint matrix and subsequently in finding the inverse of a matrix.

  3. Matrix Properties: Minors and cofactors provide insights into the properties of matrices, especially regarding invertibility and determinant properties.

Example:

Consider a 3×3 matrix A:

A=[231546072]
  1. The minor M21(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.

  2. The cofactor C21of the element a21is obtained by multiplying the minor M21 by (1)2+1because the row index + column index is odd).

Applications:

  • 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.