Chapter 3 Determinants. 3.1 The Determinant of a Matrix 3.2 Determinant and Elementary Operations 3.3 Properties of Determinants 3.4 Application of Determinants. Elementary Linear Algebra 投影片設計編製者
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Chapter 3Determinants 3.1 The Determinant of a Matrix 3.2 Determinant and Elementary Operations 3.3 Properties of Determinants 3.4 Application of Determinants Elementary Linear Algebra 投影片設計編製者 R. Larson (7 Edition) 淡江大學 電機系 翁慶昌 教授
CH 3 Linear Algebra Applied Volume of a Solid (p.108) Engineering and Control (p.124) Sudoku (p.114) Planetary Orbits (p.135)Textbook Publishing (p.137)
Minor of the entry : The determinant of the matrix determined by deleting the ith row and jth column of A • Cofactor of : Elementary Linear Algebra: Section 3.1, p.105
Notes: Sign pattern for cofactors 3 × 3 matrix 4 × 4 matrix n ×n matrix • Notes: Odd positions (where i+j is odd) have negative signs, and even positions (where i+j is even) have positive signs. Elementary Linear Algebra: Section 3.1, p.105
Thm 3.1: (Expansion by cofactors) Let A isasquare matrix of order n. Then the determinant of A is given by (Cofactor expansion along the i-th row, i=1, 2,…, n) or (Cofactor expansion along the j-th row, j=1, 2,…, n ) Elementary Linear Algebra: Section 3.1, p.107
Notes: The row (or column) containing the most zeros is the best choice for expansion by cofactors . • Ex 4: (The determinant of a matrix of order 4) Elementary Linear Algebra: Section 3.1, p.107
Sol: Elementary Linear Algebra: Section 3.1, p.107
Ex 5: –4 0 6 0 –12 16 Elementary Linear Algebra: Section 3.1, p.108
Upper triangular matrix: All the entries below the main diagonal are zeros. • Lower triangular matrix: All the entries above the main diagonal are zeros. • Diagonal matrix: All the entries above and below the main diagonal are zeros. • Note: A matrix that is both upper and lower triangular is called diagonal. Elementary Linear Algebra: Section 3.1, p.109
Thm 3.2: (Determinant of a Triangular Matrix) If A is an n × n triangular matrix (upper triangular, lower triangular, or diagonal), then its determinant is the product of the entries on the main diagonal. That is Elementary Linear Algebra: Section 3.1, p.109
Key Learning in Section 3.1 • Determine whether two matrices are equal. • Add and subtract matrices and multiply a matrix by a scalar. • Multiply two matrices. • Use matrices to solve a system of linear equations. • Partition a matrix and write a linear combination of column vectors..
Ex: Elementary Linear Algebra: Section 3.2, Addition
Notes: Elementary Linear Algebra: Section 3.2, Addition
Sol: Note: A row-echelon form of a square matrix is always upper triangular. • Ex 2: (Evaluation a determinant using elementary row operations) Elementary Linear Algebra: Section 3.2, p.113
Thm 3.4: (Conditions that yield a zero determinant) If A is a square matrix and any of the following conditions is true, then det (A) = 0. (a) An entire row (or an entire column) consists of zeros. (b) Two rows (or two columns) are equal. (c) One row (or column) is a multiple of another row (or column). Elementary Linear Algebra: Section 3.2, p.115
Ex: Elementary Linear Algebra: Section 3.2, Addition
Note: Elementary Linear Algebra: Section 3.2, p.116
Key Learning in Section 3.2 • Determine whether two matrices are equal. • Add and subtract matrices and multiply a matrix by a scalar. • Multiply two matrices. • Use matrices to solve a system of linear equations. • Partition a matrix and write a linear combination of column vectors.
3.3 Properties of Determinants • Notes: • Thm 3.5: (Determinant of a matrix product) det (AB) = det (A) det (B) (1) det (EA) = det (E) det (A) (2) (3) Elementary Linear Algebra: Section 3.3, p.120
Ex 2: • Thm 3.6: (Determinant of a scalar multiple of a matrix) If A is an n × n matrix and c is a scalar, then det (cA) = cn det (A) Find |A|. Sol: Elementary Linear Algebra: Section 3.3, p.121
Thm 3.7: (Determinant of an invertible matrix) • Ex 3: (Classifying square matrices as singular or nonsingular) A square matrix A is invertible (nonsingular) if and only if det (A) 0 Sol: A has no inverse (it is singular). B has an inverse (it is nonsingular). Elementary Linear Algebra: Section 3.3, p.122
Ex 4: • Thm 3.8: (Determinant of an inverse matrix) • Thm 3.9: (Determinant of a transpose) (a) (b) Sol: Elementary Linear Algebra: Section 3.3, pp.122-124
If A is an n × n matrix, then the following statements are equivalent. • Equivalent conditions for a nonsingular matrix: (1) A is invertible. (2) Ax = b has a unique solution for every n × 1 matrix b. (3) Ax = 0 has only the trivial solution. (4) A is row-equivalent to In (5) A can be written as the product of elementary matrices. (6) det (A) 0 Elementary Linear Algebra: Section 3.3, p.123
Sol: (a) This system does not have a unique solution. (b) This system has a unique solution. Elementary Linear Algebra: Section 3.3, p.123
Key Learning in Section 3.3 • Determine whether two matrices are equal. • Add and subtract matrices and multiply a matrix by a scalar. • Multiply two matrices. • Use matrices to solve a system of linear equations. • Partition a matrix and write a linear combination of column vectors.
