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**Chapter 3 Determinants 3.1 The Determinant of a Matrix**

3.2 Evaluation of a Determinant using Elementary Operations 3.3 Properties of Determinants 3.4 Introduction to Eigenvalues 3.5 Application of Determinants Elementary Linear Algebra 投影片設計編製者 R. Larsen et al. (6 Edition) 淡江大學 電機系 翁慶昌 教授

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**3.1 The Determinant of a Matrix**

Note: Elementary Linear Algebra: Section 3.1, p.123

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**Ex. 1: (The determinant of a matrix of order 2)**

Note: The determinant of a matrix can be positive, zero, or negative. Elementary Linear Algebra: Section 3.1, p.123

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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.124

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Ex: Elementary Linear Algebra: Section 3.1, p.124

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**Notes: Sign pattern for cofactors**

3 × 3 matrix × 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.124

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**Ex 2: Find all the minors and cofactors of A.**

Sol: (1) All the minors of A. Elementary Linear Algebra: Section 3.1, p.125

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**Sol: (2) All the cofactors of A.**

Elementary Linear Algebra: Section 3.1, p.125

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**Thm 3.1: (Expansion by cofactors) Let A is a square 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.126

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**Ex: The determinant of a matrix of order 3**

Elementary Linear Algebra: Section 3.1, Addition

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**Ex 3: The determinant of a matrix of order 3**

Sol: Elementary Linear Algebra: Section 3.1, p.126

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**Ex 5: (The determinant of a matrix of order 3)**

Sol: Elementary Linear Algebra: Section 3.1, p.128

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**The row (or column) containing the most zeros is the best choice **

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.127

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Sol: Elementary Linear Algebra: Section 3.1, p.127

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**The determinant of a matrix of order 3:**

Subtract these three products. Add these three products. Elementary Linear Algebra: Section 3.1, p.127

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Ex 5: –4 6 16 –12 Elementary Linear Algebra: Section 3.1, p.128

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**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.128

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**Ex: diagonal upper triangular lower triangular**

Elementary Linear Algebra: Section 3.1, p.128

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**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.129

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**Ex 6: Find the determinants of the following triangular matrices.**

(b) Sol: (a) |A| = (2)(–2)(1)(3) = –12 (b) |B| = (–1)(3)(2)(4)(–2) = 48 Elementary Linear Algebra: Section 3.1, p.129

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**Keywords in Section 3.1: determinant : 行列式 minor : 子行列式 cofactor : 餘因子**

expansion by cofactors : 餘因子展開 upper triangular matrix: 上三角矩陣 lower triangular matrix: 下三角矩陣 diagonal matrix: 對角矩陣

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**3.2 Evaluation of a determinant using elementary operations**

Thm 3.3: (Elementary row operations and determinants) Let A and B be square matrices. Elementary Linear Algebra: Section 3.2, p.134

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Ex: Elementary Linear Algebra: Section 3.2, Addition

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Notes: Elementary Linear Algebra: Section 3.2, pp

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**A row-echelon form of a square matrix is always upper triangular.**

Note: A row-echelon form of a square matrix is always upper triangular. Ex 2: (Evaluation a determinant using elementary row operations) Sol: Elementary Linear Algebra: Section 3.2, pp

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**Elementary Linear Algebra: Section 3.2, pp.134-135**

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Notes: Elementary Linear Algebra: Section 3.2, Addition

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**Determinants and elementary column operations**

Thm: (Elementary column operations and determinants) Let A and B be square matrices. Elementary Linear Algebra: Section 3.2, p.135

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Ex: Elementary Linear Algebra: Section 3.2, p.135

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**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.136

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Ex: Elementary Linear Algebra: Section 3.2, Addition

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**Note: Cofactor Expansion Row Reduction Order n 3 5 9 10 119 205 30 45**

Additions Multiplications 3 5 9 10 119 205 30 45 3,628,799 6,235,300 285 339 Elementary Linear Algebra: Section 3.2, p.138

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**Ex 5: (Evaluating a determinant)**

Sol: Elementary Linear Algebra: Section 3.2, p.138

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**Ex 6: (Evaluating a determinant)**

Sol: Elementary Linear Algebra: Section 3.2, p.139

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**Elementary Linear Algebra: Section 3.2, p.140**

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**3.3 Properties of Determinants**

Thm 3.5: (Determinant of a matrix product) det (AB) = det (A) det (B) Notes: (1) det (EA) = det (E) det (A) (2) (3) Elementary Linear Algebra: Section 3.3, p.143

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**Ex 1: (The determinant of a matrix product)**

Find |A|, |B|, and |AB| Sol: Elementary Linear Algebra: Section 3.3, p.143

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Check: |AB| = |A| |B| Elementary Linear Algebra: Section 3.3, p.143

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**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) Ex 2: Find |A|. Sol: Elementary Linear Algebra: Section 3.3, p.144

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**Thm 3.7: (Determinant of an invertible matrix) **

A square matrix A is invertible (nonsingular) if and only if det (A) 0 Ex 3: (Classifying square matrices as singular or nonsingular) Sol: A has no inverse (it is singular). B has an inverse (it is nonsingular). Elementary Linear Algebra: Section 3.3, p.145

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**Thm 3.8: (Determinant of an inverse matrix)**

Thm 3.9: (Determinant of a transpose) Ex 4: (a) (b) Sol: Elementary Linear Algebra: Section 3.3, pp

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**Equivalent conditions for a nonsingular matrix:**

If A is an n × n matrix, then the following statements are equivalent. (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.147

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**Ex 5: Which of the following system has a unique solution?**

(b) Elementary Linear Algebra: Section 3.3, p.148

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**This system does not have a unique solution.**

(b) This system has a unique solution. Elementary Linear Algebra: Section 3.3, p.148

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**3.4 Introduction to Eigenvalues**

Eigenvalue problem: If A is an nn matrix, do there exist n1 nonzero matrices x such that Ax is a scalar multiple of x？ Eigenvalue and eigenvector: A：an nn matrix ：a scalar x： a n1 nonzero column matrix Eigenvalue Eigenvector (The fundamental equation for the eigenvalue problem) Elementary Linear Algebra: Section 3.4, p.152

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**Ex 1: (Verifying eigenvalues and eigenvectors)**

Elementary Linear Algebra: Section 3.4, p.152

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**Given an nn matrix A, how can you find the eigenvalues and **

Question: Given an nn matrix A, how can you find the eigenvalues and corresponding eigenvectors? Characteristic equation of AMnn: If has nonzero solutions iff Note: (homogeneous system) Elementary Linear Algebra: section 3.4, p.153

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**Ex 2: (Finding eigenvalues and eigenvectors)**

Sol: Characteristic equation: Eigenvalues: Elementary Linear Algebra: Section 3.4, p.153

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**Elementary Linear Algebra: Section 3.4, p.153**

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**Ex 3: (Finding eigenvalues and eigenvectors)**

Sol: Characteristic equation: Elementary Linear Algebra: Section 3.4, p.154

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**Elementary Linear Algebra: Section 3.4, p.155**

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**Elementary Linear Algebra: Section 3.4, p.156**

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**3.5 Applications of Determinants**

Matrix of cofactors of A: Adjoint matrix of A: Elementary Linear Algebra: Section 3.5, p.158

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**Thm 3.10: (The inverse of a matrix given by its adjoint)**

If A is an n × n invertible matrix, then Ex: Elementary Linear Algebra: Section 3.5, p.159

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**(a) Find the adjoint of A.**

Ex 1 & Ex 2: (a) Find the adjoint of A. (b) Use the adjoint of A to find Sol: Elementary Linear Algebra: Section 3.5, p.160

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**adjoint matrix of A cofactor matrix of A inverse matrix of A Check:**

Elementary Linear Algebra: Section 3.5, p.160

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**(this system has a unique solution)**

Thm 3.11: (Cramer’s Rule) (this system has a unique solution) Elementary Linear Algebra: Section 3.5, p.163

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( i.e. ) Elementary Linear Algebra: Section 3.5, p.163

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Pf: A x = b, Elementary Linear Algebra: Section 3.5, p.163

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**Elementary Linear Algebra: Section 3.5, p.163**

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**Ex 4: Use Cramer’s rule to solve the system of linear equations.**

Elementary Linear Algebra: Section 3.5, p.163

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**Keywords in Section 3.5: matrix of cofactors : 餘因子矩陣**

adjoint matrix : 伴隨矩陣 Cramer’s rule : Cramer 法則

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