Lecture 3.2: Properties of Determinants

Importance of Efficient Calculation

• Determinants calculated by cofactor expansion have high operational cost (~n factorial), which is unsustainable for large matrices (e.g., 50x50 matrix).
• Incorporating row operations can significantly reduce computational cost.

Theorem 3: Properties of Determinants

1. Row Replacement:
   • alpha*Row_i + Row_j → Row_j does not change the determinant.
   • Useful for introducing zeros to reduce operational cost.
   • Caution: Avoid adding products to rows to reduce fractions as it changes the determinant.
2. Row Interchange:
   • Interchanging two rows (Row_i ↔ Row_j) introduces a factor of -1 to the determinant.
3. Row Multiplication:
   • Multiplying a row by a constant changes the determinant by that constant.
   • Factor out constants carefully.*

Combining Cofactor Expansion and Row Operations

• Effective strategy involves using both methods to compute determinants.
• Example 1:
  • Expand at a row with many zeros (e.g., Row 2).
  • Use row operations like -2*Row_1 + Row_2 → Row_2 to introduce zeros.
  • Swap rows to form an upper triangular matrix, then calculate determinant as product of diagonals.*

Important Theorems

• Invertibility and Determinant:

  • A matrix is invertible if and only if its determinant is non-zero.
  • For a 2x2 matrix, invertibility is linked to determinant being non-zero (ad - bc ≠ 0).

• Transpose and Determinant:

  • det(A^T) = det(A), supports the idea of cofactor expansion across any row/column.

• Multiplication Property:

  • det(AB) = det(A) * det(B) but not applicable to addition (det(A + B) ≠ det(A) + det(B)).
  • Example: det(A^3) = (det(A))^3.

• Inverse and Determinant:

  • det(A^(-1)) = 1/det(A) if det(A) ≠ 0.*

Practice

• Practice sheets available for applying row operations in determinant calculations.
• Understanding and applying these properties is crucial for efficient determinant computation.