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## Chapter 3 Determinants

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**Chapter 3Determinants**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 R. Larsen et al. (6 Edition)**3.1 The Determinant of a Matrix**• the determinant of a 2 × 2 matrix: • Note: Elementary Linear Algebra: Section 3.1, p.123**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**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**Ex:**Elementary Linear Algebra: Section 3.1, p.124**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.124**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**Sol: (2) All the cofactors of A.**Elementary Linear Algebra: Section 3.1, p.125**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.126**Ex: The determinant of a matrix of order 3**Elementary Linear Algebra: Section 3.1, Addition**Ex 3: The determinant of a matrix of order 3**Sol: Elementary Linear Algebra: Section 3.1, p.126**Ex 5: (The determinant of a matrix of order 3)**Sol: Elementary Linear Algebra: Section 3.1, p.128**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**Sol:**Elementary Linear Algebra: Section 3.1, p.127**The determinant of a matrix of order 3:**Subtract these three products. Add these three products. Elementary Linear Algebra: Section 3.1, p.127**Ex 5:**–4 0 6 0 –12 16 Elementary Linear Algebra: Section 3.1, p.128**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**Ex:**diagonal lower triangular upper triangular Elementary Linear Algebra: Section 3.1, p.128**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**Ex 6: Find the determinants of the following triangular**matrices. (b) (a) Sol: |A| = (2)(–2)(1)(3) = –12 (a) |B| = (–1)(3)(2)(4)(–2) = 48 (b) Elementary Linear Algebra: Section 3.1, p.129**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**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**Ex:**Elementary Linear Algebra: Section 3.2, Addition**Notes:**Elementary Linear Algebra: Section 3.2, pp.134-135**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, pp.134-135**Notes:**Elementary Linear Algebra: Section 3.2, Addition**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**Ex:**Elementary Linear Algebra: Section 3.2, p.135**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**Ex:**Elementary Linear Algebra: Section 3.2, Addition**Note:**Elementary Linear Algebra: Section 3.2, p.138**Ex 5: (Evaluating a determinant)**Sol: Elementary Linear Algebra: Section 3.2, p.138**Ex 6: (Evaluating a determinant)**Sol: Elementary Linear Algebra: Section 3.2, p.139**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.143**Ex 1: (The determinant of a matrix product)**Find |A|, |B|, and |AB| Sol: Elementary Linear Algebra: Section 3.3, p.143**Check:**|AB| = |A| |B| Elementary Linear Algebra: Section 3.3, p.143**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.144**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.145**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.146-148**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.147**Ex 5: Which of the following system has a unique solution?**(a) (b) Elementary Linear Algebra: Section 3.3, p.148**Sol:**(a) This system does not have a unique solution. (b) This system has a unique solution. Elementary Linear Algebra: Section 3.3, p.148**Eigenvalue and eigenvector:**A：an nn matrix ：a scalar x： a n1nonzero column matrix Eigenvalue Eigenvector (The fundamental equation for the eigenvalue problem) 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？ Elementary Linear Algebra: Section 3.4, p.152**Eigenvalue**Eigenvalue Eigenvector Eigenvector • Ex 1: (Verifying eigenvalues and eigenvectors) Elementary Linear Algebra: Section 3.4, p.152**Note:**(homogeneous system) If has nonzero solutions iff . • Characteristic equation of AMnn: • Question: Given an nn matrix A, how can you find the eigenvalues and corresponding eigenvectors? Elementary Linear Algebra: section 3.4, p.153**Sol: Characteristic equation:**Eigenvalues: • Ex 2: (Finding eigenvalues and eigenvectors) Elementary Linear Algebra: Section 3.4, p.153**Ex 3: (Finding eigenvalues and eigenvectors)**Sol: Characteristic equation: Elementary Linear Algebra: Section 3.4, p.154