Chapter 3 Determinants 3 1 The Determinant of

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 Application of Determinants

3. 1 The Determinant of a Matrix n the determinant of a 2 × 2 matrix: n Note: 2/62

n Ex. 1: (The determinant of a matrix of order 2) n Note: The determinant of a matrix can be positive, zero, or negative. 3/62

n n Minor of the entry : The determinant of the matrix determined by deleting the ith row and jth column of A Cofactor of : 4/62

n Ex: 5/62

n Notes: Sign pattern for cofactors 3 × 3 matrix n 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. 6/62

n Ex 2: Find all the minors and cofactors of A. Sol: (1) All the minors of A. 7/62

Sol: (2) All the cofactors of A. 8/62

n 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 ) 9/62

n Ex: The determinant of a matrix of order 3 10/62

n Ex 3: The determinant of a matrix of order 3 Sol: 11/62

n Ex 5: (The determinant of a matrix of order 3) Sol: 12/62

n n 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) 13/62

Sol: 14/62

n The determinant of a matrix of order 3: Subtract these three products. Add these three products. 15/62

n Ex 5: – 4 0 6 0 16 – 12 16/62

n Upper triangular matrix: All the entries below the main diagonal are zeros. n Lower triangular matrix: All the entries above the main diagonal are zeros. n Diagonal matrix: All the entries above and below the main diagonal are zeros. n Note: A matrix that is both upper and lower triangular is called diagonal. 17/62

n Ex: upper triangular lower triangular diagonal 18/62

n 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 19/62

n Ex 6: Find the determinants of the following triangular matrices. (a) (b) Sol: (a) |A| = (2)(– 2)(1)(3) = – 12 (b) |B| = (– 1)(3)(2)(4)(– 2) = 48 20/62

3. 2 Evaluation of a determinant using elementary operations n Thm 3. 3: (Elementary row operations and determinants) Let A and B be square matrices. 22/62

n Ex: 23/62

n Notes: 24/62

Note: A row-echelon form of a square matrix is always upper triangular. Ex 2: (Evaluation a determinant using elementary row operations) n Sol: 25/62

26/62

n Notes: 27/62

n Determinants and elementary column operations n Thm: (Elementary column operations and determinants) Let A and B be square matrices. 28/62

n Ex: 29/62

n 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). 30/62

n Ex: 31/62

n Note: Cofactor Expansion Row Reduction Order n Additions Multiplications 3 5 9 5 10 5 119 205 30 45 10 3, 628, 799 6, 235, 300 285 339 32/62

n Ex 5: (Evaluating a determinant) Sol: 33/62

n Ex 6: (Evaluating a determinant) Sol: 34/62

35/62

3. 3 Properties of Determinants n Thm 3. 5: (Determinant of a matrix product) det (AB) = det (A) det (B) n Notes: (1) det (EA) = det (E) det (A) (2) (3) 36/62

n Ex 1: (The determinant of a matrix product) Find |A|, |B|, and |AB| Sol: 37/62

n Check: |AB| = |A| |B| 38/62

Thm 3. 6: (Determinant of a scalar multiple of a matrix) n If A is an n × n matrix and c is a scalar, then det (c. A) = cn det (A) n Ex 2: Sol: Find |A|. 39/62

n Thm 3. 7: (Determinant of an invertible matrix) A square matrix A is invertible (nonsingular) if and only if det (A) 0 n Ex 3: (Classifying square matrices as singular or nonsingular) Sol: A has no inverse (it is singular). B has an inverse (it is nonsingular). 40/62

n Thm 3. 8: (Determinant of an inverse matrix) n Thm 3. 9: (Determinant of a transpose) n Ex 4: (a) (b) Sol: 41/62

n 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 42/62

n Ex 5: Which of the following system has a unique solution? (a) (b) 43/62

Sol: (a) This system does not have a unique solution. (b) This system has a unique solution. 44/62

3. 4 Introduction to Eigenvalues n Eigenvalue problem: If A is an n n matrix, do there exist n 1 nonzero matrices x such that Ax is a scalar multiple of x？ n Eigenvalue and eigenvector: A：an n n matrix ：a scalar x： a n 1 nonzero column matrix Eigenvalue (The fundamental equation for the eigenvalue problem) Eigenvector 45/62

n Ex 1: (Verifying eigenvalues and eigenvectors) Eigenvalue Eigenvector 46/62

n n Question: Given an n n matrix A, how can you find the eigenvalues and corresponding eigenvectors? Note: (homogeneous system) If n has nonzero solutions iff . Characteristic equation of A Mn n: 47/62

n Ex 2: (Finding eigenvalues and eigenvectors) Sol: Characteristic equation: Eigenvalues: 48/62

49/62

n Ex 3: (Finding eigenvalues and eigenvectors) Sol: Characteristic equation: 50/62

51/62

52/62

3. 4 Applications of Determinants n n Matrix of cofactors of A: Adjoint matrix of A: 53/62

n Thm 3. 10: (The inverse of a matrix given by its adjoint) If A is an n × n invertible matrix, then n Ex: 54/62

n Ex 1 & Ex 2: (a) Find the adjoint of A. (b) Use the adjoint of A to find Sol: 55/62

cofactor matrix of A adjoint matrix of A inverse matrix of A n Check: 56/62

n Thm 3. 11: (Cramer’s Rule) (this system has a unique solution) 57/62

( i. e. ) 58/62

n Pf: A x = b, 59/62

60/62

n Ex 4: Use Cramer’s rule to solve the system of linear equations. Sol: 61/62

Keywords in Section 3. 4: n matrix of cofactors : ﻣﺼﻔﻮﻓﺔ ﺍﻟﻤﻌﺎﻣﻼﺕ n adjoint matrix : ﻣﺼﻔﻮﻓﺔ ﻣﺼﺎﺣﺒﺔ n Cramer’s rule : ﻗﺎﻧﻮﻥ ﻛﺮﺍﻣﺮ 62/62