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Minors and Cofactors

Definition

Minor

The minor of an element in a matrix is the determinant of the smaller matrix that remains after removing the row and column containing that element. For any element \( a_{ij} \) in a matrix, the minor is denoted as \( M_{ij} \).

Cofactor

The cofactor of an element in a matrix is the minor of the element, multiplied by a sign factor. The sign is determined based on the position of the element, where the sign factor is given by \( (-1)^{i+j} \). The cofactor is denoted as \( C_{ij} \).

Mathematically:

  • Minor: \( M_{ij} \) is the determinant of the matrix obtained by deleting the \(i\)-th row and \(j\)-th column.
  • Cofactor: \( C_{ij} = (-1)^{i+j} \cdot M_{ij} \)

The cofactors are used in calculating the determinant of larger matrices and in finding the inverse of a matrix.

Steps to Find Minors and Cofactors

  1. Select an element from the matrix.
  2. Remove the row and column that contain the selected element to form a submatrix.
  3. Calculate the determinant of the submatrix. This is the minor of the element.
  4. Multiply the minor by the sign \( (-1)^{i+j} \), where \( i \) is the row number and \( j \) is the column number of the element. This gives the cofactor.

Example 1: Minors and Cofactors in a 3x3 Matrix

Let \( A \) be a 3x3 matrix: [ A = \begin{bmatrix} 1 & 2 & 3 \ 0 & 4 & 5 \ 1 & 0 & 6 \end{bmatrix} ]

Step 1: Find the minor and cofactor of element \( a_{11} \) (element in the first row, first column).

  • Element: \( a_{11} = 1 \)
  • Remove the first row and first column. The resulting submatrix is: [ \begin{bmatrix} 4 & 5 \ 0 & 6 \end{bmatrix} ]
  • The minor \( M_{11} \) is the determinant of this submatrix: [ M_{11} = (4 \cdot 6) - (5 \cdot 0) = 24 ]
  • The cofactor \( C_{11} \) is: [ C_{11} = (-1)^{1+1} \cdot M_{11} = 1 \cdot 24 = 24 ]

Step 2: Find the minor and cofactor of element \( a_{12} \) (element in the first row, second column).

  • Element: \( a_{12} = 2 \)
  • Remove the first row and second column. The resulting submatrix is: [ \begin{bmatrix} 0 & 5 \ 1 & 6 \end{bmatrix} ]
  • The minor \( M_{12} \) is the determinant of this submatrix: [ M_{12} = (0 \cdot 6) - (5 \cdot 1) = -5 ]
  • The cofactor \( C_{12} \) is: [ C_{12} = (-1)^{1+2} \cdot M_{12} = (-1) \cdot (-5) = 5 ]

Step 3: Find the minor and cofactor of element \( a_{13} \) (element in the first row, third column).

  • Element: \( a_{13} = 3 \)
  • Remove the first row and third column. The resulting submatrix is: [ \begin{bmatrix} 0 & 4 \ 1 & 0 \end{bmatrix} ]
  • The minor \( M_{13} \) is the determinant of this submatrix: [ M_{13} = (0 \cdot 0) - (4 \cdot 1) = -4 ]
  • The cofactor \( C_{13} \) is: [ C_{13} = (-1)^{1+3} \cdot M_{13} = 1 \cdot (-4) = -4 ]

Thus, the minors and cofactors of the first row of matrix \( A \) are:

  • \( M_{11} = 24 \), \( C_{11} = 24 \)
  • \( M_{12} = -5 \), \( C_{12} = 5 \)
  • \( M_{13} = -4 \), \( C_{13} = -4 \)

Example 2: Minors and Cofactors in a 2x2 Matrix

Let \( B \) be a 2x2 matrix: [ B = \begin{bmatrix} 3 & 8 \ 4 & 6 \end{bmatrix} ]

Step 1: Find the minor and cofactor of element \( b_{11} \) (element in the first row, first column).

  • Element: \( b_{11} = 3 \)
  • Remove the first row and first column. The resulting submatrix is: [ \begin{bmatrix} 6 \end{bmatrix} ]
  • The minor \( M_{11} \) is: [ M_{11} = 6 ]
  • The cofactor \( C_{11} \) is: [ C_{11} = (-1)^{1+1} \cdot M_{11} = 1 \cdot 6 = 6 ]

Step 2: Find the minor and cofactor of element \( b_{12} \) (element in the first row, second column).

  • Element: \( b_{12} = 8 \)
  • Remove the first row and second column. The resulting submatrix is: [ \begin{bmatrix} 4 \end{bmatrix} ]
  • The minor \( M_{12} \) is: [ M_{12} = 4 ]
  • The cofactor \( C_{12} \) is: [ C_{12} = (-1)^{1+2} \cdot M_{12} = (-1) \cdot 4 = -4 ]

Thus, the minors and cofactors of matrix \( B \) are:

  • \( M_{11} = 6 \), \( C_{11} = 6 \)
  • \( M_{12} = 4 \), \( C_{12} = -4 \)

Conclusion

  • The minor of an element is the determinant of the submatrix formed by deleting the row and column of that element.
  • The cofactor is the minor of an element multiplied by a sign factor \( (-1)^{i+j} \), depending on its position in the matrix.
  • Minors and cofactors are essential in calculating the determinant of larger matrices and finding the inverse of a matrix.
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