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Chap. 3 Determinants. 3.1 The Determinants of a Matrix 3.2 Evaluation of a Determinant Using Elementary Operations 3.3 Properties of Determinants 3.4 Introduction to Eigenvalues 3.5 Applications of Determinants. . +. 3.1 The Determinant of a Matrix.

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## Chap. 3 Determinants

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**Chap. 3Determinants**3.1 The Determinants of a Matrix 3.2 Evaluation of a Determinant Using Elementary Operations 3.3 Properties of Determinants 3.4 Introduction to Eigenvalues 3.5 Applications of Determinants****+ 3.1 The Determinant of a Matrix • Every square matrix can be associated with a real number called its determinant. • Definition: The determinant of the matrixis given by • Example 1: 2 2(2) 1(3) = 7 2(2) 1(4) = 0 0(4) 2(3) = 6 Chapter 3**Section 3-1**Minors and Cofactors of a Matrix • If A is a square matrix, then the minor (子行列式) Mij of the element aij is the determinant of the matrix obtained by deleting the ith row and jth column of A.The cofactor (餘因子) Cijis given by Cij = (1)i+jMij. • Sign pattern for cofactors: Chapter 3**Section 3-1** + + + Theorem 3.1 Expansion by Cofactors • Let A be a square matrix of order n. Then the determinant of A is given by • For any 33 matrix: ith row expansion jth column expansion Chapter 3**Section 3-1**Examples 2 & 3 • Find all the minors and cofactors of A, and then find the determinant of A. Sol: Chapter 3**Section 3-1**(4) (6) (0) +(16) +(0) +(12) Example 5 • Find the determinant of Sol: Chapter 3**Section 3-1**Example 4 • Find the determinant of Sol: Expansion by which row or which column? the 3rd column: three of the entires are zeros Chapter 3**Section 3-1**Triangular Matrices Upper triangular Matrix Lower triangular Matrix • Theorem 3.2: If A is a triangular matrix of order n, then its determinant is the product of the entires on the main diagonal. That is, Chapter 3**Section 3-1**Example Chapter 3**3.2 Evaluation of a Determinant Using Elementary**Operations • Which of the following two determinants is easier to evaluate? By elementary row operations Chapter 3**Section 3-2**(2) Theorem 3.3 Elementary Row Operations and Determinants • Let A and B be square matrices. 1. If B is obtained from A by interchanging two rows of A, then det(B) = det(A). 2. If B is obtained from A by adding a multiple of a row of Ato another row of A, then det(B) = det(A). 3. If B is obtained from A by multiplying a row of Aby a nonzero constant c, then det(B) = cdet(A). Take a common factor out of a row 3 Chapter 3**Section 3-2**(2) Example 2 • Find the determinant of Sol: Factor 7 out of the 2nd row (1) Chapter 3**Section 3-2**Determinants andElementary Column Operations • Although Theorem 3.3 was stated in terms of elementary row operations, the theorem remains validif the word “row” is replaced by the word “column.” • Operations performed on the column of a matrix are called elementary column operations. • Two matrices are called column-equivalent if one can be obtained from the other by elementary column operations. Chapter 3**Section 3-2**Expansion by the second column (2) Example 3 • Find the determinant of Sol: Chapter 3**Section 3-2**(3) (1) Theorem 3.4 Conditions That Yield a Zero Determinant • If A is a square matrix and any one of the following conditions is true, then det(A) = 0. 1. Anentire row (or an entire column) consists of zeros. 2. Two rows (or columns) are equal. 3. One row (or column) is a multiple of another row (or column). Chapter 3**Section 3-2**(2) Examples 4 & 5 (2) Chapter 3**Section 3-2**(1) (3) Example 6 • Find the determinant of Sol: Chapter 3**3.3 Properties of Determinants**• Example 1: Find for the matrices Sol: Chapter 3**Section 3-3**Theorems 3.5 & 3.6 Theorem 3.5: Determinant of a Matrix Product • If A and B are square matrices of order n, thendet(AB) = det(A) det(B) Remark: Theorem 3.6: Determinant of a Scalar Multiple of a Matrix • If A is a nn matrix and c is a scalar, then the determinant of cA is given by det(cA) = cn det(A). Remark: [Thm. 3.3]If B is obtained from A by multiplying a row of Aby a nonzero constant c, then det(B) = cdet(A). Chapter 3**Section 3-3**Example 2 • Find the determinant of the matrix Sol: Chapter 3**Section 3-3** Theorems 3.7 & 3.8 Theorem 3.7: Determinant of an Invertible Matrix • A square matrix A is invertible (nonsingular) if and only if det(A) 0. Theorem 3.8: Determinant of an Inverse Matrix • If A is invertible, then det(A1) = 1 / det(A). Hint: A is invertible AA1 = I Chapter 3**Section 3-3**Example 3 & 4 Example 3: Which of the matrices has an inverse? Sol: Example 4: Find for the matrix Sol: It has an inverse. It has no inverse. Chapter 3**Section 3-3** Equivalent Conditions for a Nonsingular Matrix • If A is an nn matrix, then the following statements are equivalent. 1. A is invertible. 2. Ax = b has a unique solution for everyn1 column vector b. 3. Ax = O has only the trivial solution. 4. A is row-equivalent to In. 5. A can be written as the product of elementary matrices. 【 Also see in Theorem 2.15 】 6. det(A) 0. 【 See Example 5 (p.148) for instance 】 Chapter 3**Section 3-3**Determinant of a Transpose Theorem 3.9: If A is a square matrix, then det(A)=det(AT). Example 6: Show that for the following matrix. pf: Chapter 3**3.4 Introduction to Eigenvalues**See Chapter 7 Chapter 3**3.5 Applications of Determinants**• The Adjoint of a MatrixIf A is a square matrix, then the matrix of cofactors of Ahas the form • The transpose of this matrixis called the adjoint of A andis denoted by adj(A). Chapter 3**Section 3-5**Example 1 • Find the adjoint of Sol:The matrix of cofactors of A: Chapter 3**Section 3-5** Theorem 3.10 The Inverse of a Matrix Given by Its Adjoint • If A is an nninvertible matrix, then • If A is 22 matrixthen the adjoint of A is .Form Theorem 3.10 you have Chapter 3**Section 3-5**Example 2 • Use the adjoint of to find . Sol: Chapter 3**Section 3-5**Theorem 3.11: Cramer’s Rule • If a system of n linear equations in n variables has a coefficient matrix with a nonzero determinant ,then the solution of the system is given bywhere the ith column of Ai is the column of constants in the system of equations. Chapter 3**Section 3-5**Example 4 • Use Cramer’s Rule to solve the system of linear equationfor x. Sol: Chapter 3**Section 3-5**Area of a Triangle • The area of a triangle whose verticesare (x1, y1), (x2, y2), and (x3, y3) isgiven bywhere the sign () is chosen to give a positive area. pf: Area = Chapter 3**Section 3-5**(4,3) (2,2) Example 5 Fine the area of the triangle whose vertices are (1, 0), (2, 2), and (4, 3). Sol: • Fine the area of the triangle whosevertices are (0, 1), (2, 2), and (4, 3). (1,0) Three points in the xy-plane lie on the same line. Chapter 3**Section 3-5**Collinear Pts & Line Equation • Test for Collinear Points in the xy-PlaneThree points (x1, y1), (x2, y2), and (x3, y3) are collinearif and only if • Two-Point Form of the Equation of a LineAn equation of the line passing through the distinct points (x1, y1) and (x2, y2) is given by The 3rd point: (x, y) Chapter 3**Section 3-5**Example 6 • Find an equation of the line passing through the points(2, 4) and (1, 3). Sol: An equation of the line is x 3y = 10. Chapter 3**The volume of the tetrahedron whose verticesare (x1,y1, z1),**(x2, y2, z2), (x3, y3, z3), and (x4, y4, z4), is given by where the sign () is chosen to give a positive area. Example 7: Find the volume of the tetrahedron whose vertices are (0,4,1), (4,0,0), (3,5,2), and (2,2,5). Sol: Section 3-5 Volume of Tetrahedron Chapter 3**Test for Coplanar Points in SpaceFour points (x1,y1, z1),**(x2, y2, z2), (x3, y3, z3), and (x4, y4, z4) are coplanar if and only if Three-Point Form of the Equation of a PlaneAn equation of the plane passing through the distinct points (x1,y1, z1), (x2, y2, z2), and (x3, y3, z3) is given by Section 3-5 Coplanar Pts & Plane Equation Chapter 3**Section 3-5**(1) Example 8 • Find an equation of the plane passing through the points(0,1,0), (1,3,2) and (2,0,1). Sol: Chapter 3

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