Evaluation of a Determinant

Evaluating the determinant of a matrix involves various methods, including expansion by minors, cofactor expansion, and using properties of determinants. The most common methods for evaluating determinants are:

1. Expansion by Minors or Cofactors:

  • Method: Select a row or column, expand the determinant by minors or cofactors along that row or column.

  • Formula: For a square matrix A of order n:

    det(A)=j=1naij×Cijordet(A)=j=1naij×(1)i+j×Mij

    where Cij is the cofactor of aij and Mij is the minor of aij.

2. Row or Column Operations:

  • Method: Apply elementary row or column operations to transform the matrix into a triangular or diagonal form.

  • Formula: For an upper triangular or lower triangular matrix, the determinant is the product of the diagonal elements.

3. Properties of Determinants:

  • Method: Utilize determinant properties to simplify the calculation or transform the matrix.

  • Formula: Apply properties such as transposition, scaling rows or columns, or combining rows or columns to simplify the determinant calculation.

Example:

Consider a 3×3 matrix A:

A=[231546072]

To evaluate det(A):

  • One method is to expand along the first row:
    det(A)=2×det([4672])3×det([5602])+1×det([5407])
  • Another approach could be using row operations to convert A into upper triangular form and then calculate the determinant as the product of the diagonal elements.

Sarrus' Rule:

For a 3×3 matrix:

A=[abcdefghi]

To find det(A) using Sarrus' rule:

  1. Extend the Matrix:
    • Write the first two columns of the matrix to the right of the original matrix.
[abcdefghi]abdegh
  1. Multiply Diagonals:

    • Multiply the elements along the three diagonals going from the upper left to the lower right.
  2. Sum the Products:

    • Add the products obtained in step 2 moving from left to right.
    • Subtract the sum of the products obtained by moving from right to left.

Example:

Consider the matrix:

A=[235114023]

Applying Sarrus' rule:

det(A)=(2×(1)×(3))+(3×4×0)+(5×1×2)(0×(1)×5)(2×4×2)((3)×1×2)
det(A)=(6)+(0)+(10)(0)(16)(6)=6+1016+6=6

Significance:

  • Sarrus' rule simplifies the calculation of 3×3 determinants by following a specific pattern without needing to compute minors or cofactors.

  • This method is useful for relatively smaller matrices where applying direct computation by expanding determinants might be more time-consuming.