Learning Objectives
- Understand determinants and their evaluation
- Learn properties of determinants
- Study cofactors, adjoint, and inverse of a matrix
- Apply Cramer's rule to solve linear equations
Key Concepts
Determinant Evaluation
2×2 matrix: |A| = a₁₁a₂₂ - a₁₂a₂₁. 3×3 matrix: Expand along any row or column using cofactors. |A| = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ (expansion along R1). Minor Mij: Determinant of matrix obtained by deleting row i and column j. Cofactor Cij = (-1)i+j Mij.
Properties of Determinants
(1) |AT| = |A|. (2) Interchanging two rows/columns changes sign. (3) If two rows/columns are identical, |A| = 0. (4) Multiplying a row/column by k multiplies determinant by k. (5) |kA| = kn|A| for n×n matrix. (6) If elements of a row/column are expressed as sum, determinant can be split. (7) Adding multiple of one row to another doesn't change value. (8) |AB| = |A||B|. (9) |A-1| = 1/|A|. (10) |An| = |A|n.
Adjoint and Inverse
Adjoint (adj A): Transpose of cofactor matrix. A(adj A) = (adj A)A = |A|I. Inverse: A-1 = (adj A)/|A| (exists only if |A| ≠ 0 — non-singular matrix). If |A| = 0, matrix is singular (no inverse).
Properties of adjoint: |adj A| = |A|n-1 (for n×n matrix). adj(AB) = (adj B)(adj A). adj(AT) = (adj A)T.
System of Linear Equations (Cramer's Rule)
For system AX = B where A is coefficient matrix: |A| ≠ 0: Unique solution X = A-1B. Using Cramer's rule: x = D₁/D, y = D₂/D, z = D₃/D where D = |A| and D₁, D₂, D₃ are obtained by replacing respective columns with B.
|A| = 0 and (adj A)B = O: Infinitely many solutions (consistent). |A| = 0 and (adj A)B ≠ O: No solution (inconsistent).
Area of Triangle using Determinant
Area = (1/2)|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)| = (1/2)|det[x₁ y₁ 1; x₂ y₂ 1; x₃ y₃ 1]|. Points are collinear if area = 0.
Summary
Determinants are scalar values associated with square matrices. They have important properties regarding row/column operations. The adjoint and inverse are computed using cofactors. Cramer's rule uses determinants to solve systems of linear equations. The nature of solutions depends on whether the determinant is zero or non-zero.
Important Terms
- Minor: Determinant of submatrix obtained by deleting a row and column
- Cofactor: Signed minor; Cij = (-1)i+jMij
- Adjoint: Transpose of cofactor matrix
- Singular matrix: Matrix with determinant = 0 (no inverse)
- Cramer's rule: Solving linear equations using determinant ratios
Quick Revision
- |AT| = |A|; |AB| = |A||B|; |kA| = kn|A|
- A-1 = adj(A)/|A|; exists only if |A| ≠ 0
- |adj A| = |A|n-1 for n×n matrix
- Identical rows/columns ⟹ |A| = 0
- Row interchange changes sign of determinant
- Cramer's: x = D₁/D, y = D₂/D, z = D₃/D
- |A| ≠ 0 → unique solution; |A| = 0 → check adjoint