1. 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

2. 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
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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:
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4. 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    + + +
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5. Section 3-1
Examples 2 & 3
Find all the minors and cofactors of A, and then find the determinant of A.
Sol:
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6. 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)
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7. 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
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8. 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,
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9. Section 3-1
Example
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10. 3.2 Evaluation of a Determinant Using Elementary Operations
Which of the following two determinants is easier to evaluate?
By elementary row operations
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11. 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)
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12. 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)
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13. 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.
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14. Expansion by the second column
Section 3-2
Example 3
Find the determinant of
Sol:
Expansion by the second column
(2)
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15.  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)
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16. Section 3-2
Examples 4 & 5
(2)
(2)
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17. Example 6 Find the determinant of Sol: (1) (3) Section 3-2
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18. 3.3 Properties of Determinants
Example 1: Find for the matrices
Sol:
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19. 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).
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20. Example 2 Find the determinant of the matrix Sol: Section 3-3
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21.  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
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22. 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.
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23.  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 】
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24. 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:
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25. 3.4 Introduction to Eigenvalues
See Chapter 7
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26. 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).
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27. 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:
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28.  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
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29. Example 2 Use the adjoint of to find . Sol: Section 3-5 Ming-Feng Yeh
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30. 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.
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31. Section 3-5
Example 4
Use Cramer’s Rule to solve the system of linear equation for x.
Sol:
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32. 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 =
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33. 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.
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34. 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)
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35. 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.
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36. 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:
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37. 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
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38. 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)
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