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 Every square matrix can be associated with a real number called its determinant. Definition: The determinant of the matrix is given by Example 1:  + 2(2)  1(3) = 7 2 2(2)  1(4) = 0 0(4)  2(3) = 6 Ming-Feng Yeh Chapter 3

Minors and Cofactors of a Matrix 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) Cij is given by Cij = (1)i+jMij. Sign pattern for cofactors: Ming-Feng Yeh Chapter 3

Theorem 3.1 Expansion by Cofactors 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 33 matrix: ith row expansion jth column expansion    + + + Ming-Feng Yeh Chapter 3

Section 3-1 Examples 2 & 3 Find all the minors and cofactors of A, and then find the determinant of A. Sol: Ming-Feng Yeh Chapter 3

Example 5 Find the determinant of Sol: (4) (0) (6) +(0) +(16) Section 3-1 Example 5 Find the determinant of Sol: (4) (0) (6) +(0) +(16) +(12) Ming-Feng Yeh Chapter 3

Example 4 Find the determinant of 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 Ming-Feng Yeh Chapter 3

Triangular Matrices Upper triangular Matrix Lower triangular Matrix 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, Ming-Feng Yeh Chapter 3

Section 3-1 Example Ming-Feng Yeh 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 Ming-Feng Yeh Chapter 3

Take a common factor out of a row Section 3-2  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 A to another row of A, then det(B) = det(A). 3. If B is obtained from A by multiplying a row of A by a nonzero constant c, then det(B) = cdet(A). Take a common factor out of a row  3 (2) Ming-Feng Yeh Chapter 3

Example 2 Find the determinant of Sol: (2) Section 3-2 Example 2 Find the determinant of Sol: (2) Factor 7 out of the 2nd row (1) Ming-Feng Yeh Chapter 3

Determinants and Elementary Column Operations Section 3-2 Determinants and Elementary Column Operations Although Theorem 3.3 was stated in terms of elementary row operations, the theorem remains valid if 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. Ming-Feng Yeh Chapter 3

Expansion by the second column Section 3-2 Example 3 Find the determinant of Sol: Expansion by the second column (2) Ming-Feng Yeh Chapter 3

 Theorem 3.4 Conditions That Yield a Zero Determinant Section 3-2  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. An entire 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). (3) (1) Ming-Feng Yeh Chapter 3

Section 3-2 Examples 4 & 5 (2) (2) Ming-Feng Yeh Chapter 3

Example 6 Find the determinant of Sol: (1) (3) Section 3-2 Ming-Feng Yeh Chapter 3

3.3 Properties of Determinants Example 1: Find for the matrices Sol: Ming-Feng Yeh Chapter 3

Theorems 3.5 & 3.6 Theorem 3.5: Determinant of a Matrix Product 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, then det(AB) = det(A) det(B) Remark: Theorem 3.6: Determinant of a Scalar Multiple of a Matrix If A is a nn 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 A by a nonzero constant c, then det(B) = cdet(A). Ming-Feng Yeh Chapter 3

Example 2 Find the determinant of the matrix Sol: Section 3-3 Ming-Feng Yeh Chapter 3

 Theorems 3.7 & 3.8 Theorem 3.7: Determinant of an Invertible Matrix 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(A1) = 1 / det(A). Hint: A is invertible  AA1 = I Ming-Feng Yeh Chapter 3

Example 3 & 4 Example 3: Which of the matrices has an inverse? Sol: Section 3-3 Example 3 & 4 Example 3: Which of the matrices has an inverse? Sol: Example 4: Find for the matrix It has no inverse. It has an inverse. Ming-Feng Yeh Chapter 3

 Equivalent Conditions for a Nonsingular Matrix Section 3-3  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 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 】 Ming-Feng Yeh Chapter 3

Determinant of a Transpose 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: Ming-Feng Yeh Chapter 3

3.4 Introduction to Eigenvalues See Chapter 7 Ming-Feng Yeh Chapter 3

3.5 Applications of Determinants The Adjoint of a Matrix If A is a square matrix, then the matrix of cofactors of A has the form The transpose of this matrix is called the adjoint of A and is denoted by adj(A). Ming-Feng Yeh Chapter 3

Example 1 Find the adjoint of Sol: The matrix of cofactors of A: Section 3-5 Example 1 Find the adjoint of Sol: The matrix of cofactors of A: Ming-Feng Yeh Chapter 3

 Theorem 3.10 The Inverse of a Matrix Given by Its Adjoint Section 3-5  Theorem 3.10 The Inverse of a Matrix Given by Its Adjoint If A is an nn invertible matrix, then If A is 22 matrix then the adjoint of A is . Form Theorem 3.10 you have Ming-Feng Yeh Chapter 3

Example 2 Use the adjoint of to find . Sol: Section 3-5 Ming-Feng Yeh Chapter 3

Theorem 3.11: Cramer’s Rule 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 by where the ith column of Ai is the column of constants in the system of equations. Ming-Feng Yeh Chapter 3

Section 3-5 Example 4 Use Cramer’s Rule to solve the system of linear equation for x. Sol: Ming-Feng Yeh Chapter 3

Section 3-5 Area of a Triangle The area of a triangle whose vertices are (x1, y1), (x2, y2), and (x3, y3) is given by where the sign () is chosen to give a positive area. pf: Area = Ming-Feng Yeh Chapter 3

Section 3-5 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 whose vertices are (0, 1), (2, 2), and (4, 3). (2,2) (4,3) (1,0) Three points in the xy-plane lie on the same line. Ming-Feng Yeh Chapter 3

Collinear Pts & Line Equation Section 3-5 Collinear Pts & Line Equation Test for Collinear Points in the xy-Plane Three points (x1, y1), (x2, y2), and (x3, y3) are collinear if and only if Two-Point Form of the Equation of a Line An equation of the line passing through the distinct points (x1, y1) and (x2, y2) is given by The 3rd point: (x, y) Ming-Feng Yeh 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. Ming-Feng Yeh Chapter 3

Section 3-5 Volume of Tetrahedron The volume of the tetrahedron whose vertices are (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: Ming-Feng Yeh Chapter 3

Coplanar Pts & Plane Equation Section 3-5 Coplanar Pts & Plane Equation Test for Coplanar Points in Space Four 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 Plane An equation of the plane passing through the distinct points (x1,y1, z1), (x2, y2, z2), and (x3, y3, z3) is given by Ming-Feng Yeh Chapter 3

Section 3-5 Example 8 Find an equation of the plane passing through the points (0,1,0), (1,3,2) and (2,0,1). Sol: (1) Ming-Feng Yeh Chapter 3