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CHAPTER 3�DETERMINANTS

Elementary Linear Algebra

R. Larson (8 Edition)

3.1 The Determinant of a Matrix

3.2 Determinant and Elementary Operations

3.3 Properties of Determinants

3.4 Application of Determinants

投影片設計製作者

淡江大學電機系翁慶昌教授

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CH 3 Linear Algebra Applied

Volume of a Tetrahedron (p.114) Engineering and Control (p.130)

Sudoku (p.120)

Comet Landing (p.141) Software Publishing (p.143)

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3.1 The Determinant of a Matrix

the determinant of a 2 × 2 matrix:

Note:

Elementary Linear Algebra: Section 3.1, p.110

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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.110

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Cofactor of :

Minor of the entry :

The determinant of the matrix determined by deleting the ith row and jth column of A

Elementary Linear Algebra: Section 3.1, p.111

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Elementary Linear Algebra: Section 3.1, p.111

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Notes: Sign pattern for cofactors

Elementary Linear Algebra: Section 3.1, p.111

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.

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Ex 2: Find all the minors and cofactors of A.

Sol: (1) All the minors of A.

Elementary Linear Algebra: Section 3.1, p.111

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Elementary Linear Algebra: Section 3.1, p.111

Sol: (2) All the cofactors of A.

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Thm 3.1: (Expansion by cofactors)

(Cofactor expansion along the i-th row, i =1, 2,…, n )

(Cofactor expansion along the j-th row, j =1, 2,…, n )

Let A is a square matrix of order n.

Then the determinant of A is given by

Elementary Linear Algebra: Section 3.1, p.113

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Ex: The determinant of a matrix of order 3

Elementary Linear Algebra: Section 3.1, Addition

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Ex 3: The determinant of a matrix of order 3

Elementary Linear Algebra: Section 3.1, p.112

Sol:

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Ex 5: (The determinant of a matrix of order 3)

Sol:

Elementary Linear Algebra: Section 3.1, p.114

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Ex 4: (The determinant of a matrix of order 4)

Elementary Linear Algebra: Section 3.1, p.113

Notes:

The row (or column) containing the most zeros is the best choice

for expansion by cofactors .

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Sol:

Elementary Linear Algebra: Section 3.1, p.113

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The determinant of a matrix of order 3:

Add these three products.

Subtract these three products.

Elementary Linear Algebra: Section 3.1, p.114

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Ex 5:

–4

–12

Elementary Linear Algebra: Section 3.1, p.114

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Upper triangular matrix:

Lower triangular matrix:

Diagonal matrix:

All the entries below the main diagonal are zeros.

All the entries above the main diagonal are zeros.

All the entries above and below the main diagonal are zeros.

Elementary Linear Algebra: Section 3.1, p.115

Note:

A matrix that is both upper and lower triangular is called diagonal.

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upper triangular

lower triangular

diagonal

Elementary Linear Algebra: Section 3.1, p.115

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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.115

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Ex 6: Find the determinants of the following triangular matrices.

(a)

(b)

|A| = (2)(–2)(1)(3) = –12

|B| = (–1)(3)(2)(4)(–2) = 48

(a)

(b)

Sol:

Elementary Linear Algebra: Section 3.1, p.115

老師，這個例題只有(a)小題，請確認是否需要修改

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Key Learning in Section 3.1

Find the determinant of a 2 × 2 matrix.
Find the minors and cofactors of a matrix.
Use expansion by cofactors to find the determinant of a matrix.
Find the determinant of a triangular matrix.

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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

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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.119

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Elementary Linear Algebra: Section 3.2, Addition

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Notes:

Elementary Linear Algebra: Section 3.2, Addition

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Note:

A row-echelon form of a square matrix is always upper triangular.

Ex 2: (Evaluation a determinant using elementary row operations)

Sol:

Elementary Linear Algebra: Section 3.2, p.119

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Elementary Linear Algebra: Section 3.2, p.119

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Notes:

Elementary Linear Algebra: Section 3.2, Addition

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Determinants and elementary column operations

Elementary Linear Algebra: Section 3.2, p.120

Let A and B be square matrices.

Thm: (Elementary column operations and determinants)

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Elementary Linear Algebra: Section 3.2, p.120

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Thm 3.4: (Conditions that yield a zero determinant)

(a) An entire row (or an entire column) consists of zeros.

(b) Two rows (or two columns) are equal.

If A is a square matrix and any of the following conditions is true, then det (A) = 0.

Elementary Linear Algebra: Section 3.2, p.121

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Elementary Linear Algebra: Section 3.2, Addition

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

Note:

Elementary Linear Algebra: Section 3.2, p.122

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Ex 5: (Evaluating a determinant)

Sol:

Elementary Linear Algebra: Section 3.2, p.122

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Ex 6: (Evaluating a determinant)

Sol:

Elementary Linear Algebra: Section 3.2, p.123

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Elementary Linear Algebra: Section 3.2, p.123

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Key Learning in Section 3.2

Use elementary row operations to evaluate a determinant.
Use elementary column operations to evaluate a determinant.
Recognize conditions that yield zero determinants.

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Keywords in Section 3.2

determinant : 行列式
elementary row operation: 基本列運算
row equivalent: 列等價
elementary matrix: 基本矩陣
elementary column operation: 基本行運算
column equivalent: 行等價

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3.3 Properties of Determinants

Notes:

Thm 3.5: (Determinant of a matrix product)

(1) det(EA) = det(E) det(A)

(2)

(3)

det (AB) = det (A) det (B)

Elementary Linear Algebra: Section 3.3, p.126

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Ex 1: (The determinant of a matrix product)

Sol:

Find |A|, |B|, and |AB|

Elementary Linear Algebra: Section 3.3, p.126

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|AB| = |A| |B|

Check:

Elementary Linear Algebra: Section 3.3, p.126

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Ex 2:

Find |A|.

Sol:

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)

Elementary Linear Algebra: Section 3.3, p.127

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Ex 3: (Classifying square matrices as singular or nonsingular)

A has no inverse (it is singular).

B has an inverse (it is nonsingular).

Sol:

Thm 3.7: (Determinant of an invertible matrix)

A square matrix A is invertible (nonsingular) if and only if

det (A) ≠ 0

Elementary Linear Algebra: Section 3.3, p.128

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Ex 4:

(a)

(b)

Sol:

Thm 3.8: (Determinant of an inverse matrix)

Thm 3.9: (Determinant of a transpose)

Elementary Linear Algebra: Section 3.3, pp.128-130

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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 I_n

(5) A can be written as the product of elementary matrices.

(6) det (A) ≠ 0

Equivalent conditions for a nonsingular matrix:

Elementary Linear Algebra: Section 3.3, p.129

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Ex 5: Which of the following system has a unique solution?

(a)

(b)

Elementary Linear Algebra: Section 3.3, p.129

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Sol:

(a)

This system does not have a unique solution.

(b)

This system has a unique solution.

Elementary Linear Algebra: Section 3.3, p.129

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Key Learning in Section 3.3

Find the determinant of a matrix product and a scalar multiple of a matrix.
Find the determinant of an inverse matrix and recognize equivalent conditions for a nonsingular matrix.
Find the determinant of the transpose of a matrix.

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Keywords in Section 3.3

determinant: 行列式
matrix multiplication: 矩陣相乘
scalar multiplication: 純量積
invertible matrix: 可逆矩陣
inverse matrix: 反矩陣
nonsingular matrix: 非奇異矩陣
transpose matrix: 轉置矩陣

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3.4 Applications of Determinants

Matrix of cofactors of A:

Adjoint matrix of A:

Elementary Linear Algebra: Section 3.4, p.134

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Thm 3.10: (The inverse of a matrix given by its adjoint)

If A is an n × n invertible matrix, then

Elementary Linear Algebra: Section 3.4, p.135

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Ex 1 & Ex 2:

(a) Find the adjoint of A.

(b) Use the adjoint of A to find

Sol:

Elementary Linear Algebra: Section 3.4, pp.134-135

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cofactor matrix of A

adjoint matrix of A

inverse matrix of A

Check:

Elementary Linear Algebra: Section 3.4, pp.134-135

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Cramer’s Rule

Cramer’s Rule uses determinants to solve a system of linear equations in variables. This rule applies only to systems with unique solutions.

Elementary Linear Algebra: Section 3.4, pp.136

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Ex 3: (Using Cramer’s Rule)

Sol: Find the determinant of the coefficient matrix

Use Cramer’s Rule to solve the system of linear equations.

Elementary Linear Algebra: Section 3.4, pp.136

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Thm 3.11: (Cramer’s Rule)

(this system has a unique solution)

Elementary Linear Algebra: Section 3.4, p.137

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( i.e.

)

Elementary Linear Algebra: Section 3.4, p.137

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Pf:

Ax = b,

Elementary Linear Algebra: Section 3.4, p.137

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Elementary Linear Algebra: Section 3.4, p.137

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Ex 4: Use Cramer’s rule to solve the system of linear equations.

Sol:

Elementary Linear Algebra: Section 3.4, p.137

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Area of a triangle in the xy-plane:

A triangle with vertices

where the sign (±) is chosen to give a positive area.

Pf:

Consider the three trapezoid

Trapezoid 1: (x₁, 0), (x₁, y₁), (x₃, y₃), (x₃, 0)

Trapezoid 2: (x₃, 0), (x₃, y₃), (x₂, y₂), (x₂, 0)

Trapezoid 3: (x₁, 0), (x₁, y₁), (x₂, y₂), (x₂, 0)

Elementary Linear Algebra: Section 3.4, p.138

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If the vertices do not occur in the order x₁≦x₂≦x₃ or if the vertex (x₃, y₃) is not above the line segment connecting the other two vertices, then the formula above may yield the negative of the area. So, use ± and choose the correct sign to give a positive area.

Elementary Linear Algebra: Section 3.4, p.138

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Ex 5: (Finding the Area of a Triangle)

Sol:

The area of the triangle is square units.

If three points in the xy-plane lie on the same line, then the determinant in the formula for the area of a triangle is zero.

Find the area of the triangle whose vertices are

(1, 1), (2, 2), and (4, 3).

Elementary Linear Algebra: Section 3.4, p.138-139

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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

Elementary Linear Algebra: Section 3.4, p.139

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Ex 6: (Finding an Equation of the Line Passing Through Two Points)

Sol:

Find an equation of the line passing through the points

(2, 4) and (-1, 3).

Elementary Linear Algebra: Section 3.4, p.139

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Volume of a Tetrahedron:

The volume of a tetrahedron with vertices (x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃), and (x₄, y₄, z₄) is

where the sign (±) is chosen to give a positive area.

Elementary Linear Algebra: Section 3.4, p.140

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Ex 7: (Finding the Volume of a Tetrahedron)

Sol:

The volume of the tetrahedron is 12 cubic units.

Find the volume of the tetrahedron shown in the following figure, whose vertices are (0, 4, 1), (4, 0, 0), (3, 5, 2), and (2, 2, 5).

Elementary Linear Algebra: Section 3.4, p.140

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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 line:

An equation of the line passing through the distinct points

(x₁, y₁, z₁), (x₂, y₂, z₂), and (x₃, y₃, z₃) is given by

Elementary Linear Algebra: Section 3.4, p.140

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Ex 8: (Finding an Equation of the Plane Passing Through Three Points)

Sol:

Find an equation of the plane passing through the points (0, 1, 0), (−1, 3, 2), and (-2, 0, 1).

Elementary Linear Algebra: Section 3.4, p.141

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Key Learning in Section 3.4

Find the adjoint of a matrix and use it to find the inverse of the matrix.
Use Cramer’s Rule to solve a system of n linear equations in n variables.
Use determinants to find area, volume, and the equations of lines and planes.

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Keywords in Section 3.4

matrix of cofactors : 餘因子矩陣
adjoint matrix : 伴隨矩陣
Cramer’s rule : 克萊姆法則
area: 面積
volume: 體積
triangle: 三角形
tetrahedron: 四面體

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Volume of a Tetrahedron

Recall that a tetrahedron is a polyhedron consisting of four triangular faces. One practical application of determinants is in finding the volume of a tetrahedron in a coordinate plane. If the vertices of a tetrahedron are (x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃), and (x₄, y₄, z₄), then the volume is

You will study this and other applications of determinants in Section 3.4.

3.1 Linear Algebra Applied

Elementary Linear Algebra: Section 3.1, p.114

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Sudoku

In a Sudoku puzzle, the object is to fill out a partially completed 9🞨9 grid of boxes with numbers from 1 to 9 so that each column, row, and 3🞨3 sub-grid contains each of these numbers without repetition. For a completed Sudoku grid to be valid, no two rows (or columns) will have the numbers in the same order. If this should happen, then the determinant of the matrix formed by the numbers will be zero. This is a direct result of condition 2 of Theorem 3.4.

3.2 Linear Algebra Applied

Elementary Linear Algebra: Section 3.2, p.120

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Engineering and Control

Systems of linear differential equations often arise in engineering and control theory. For a function f(t) that is defined for all positive values of t, the Laplace transform of f(t) is

provided that the improper integral exists. Laplace transforms and Cramer’s Rule, which uses determinants to solve a system of linear equations, can sometimes be used to solve a system of differential equations. You will study Cramer’s Rule in the next section.

3.3 Linear Algebra Applied

Elementary Linear Algebra: Section 3.3, p.130

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Comet Landing

On November 12, 2014, European Space Agency’s Rosetta orbiting spacecraft landed the probe Philae on the surface of the comet 67P/Churyumov-Gerasimenko. Comets that orbit the Sun, such as 67P, follow Kepler’s First Law of Planetary Motion. This law states that the orbit is an ellipse, with the sun at one focus of the ellipse. The general equation of a conic section, such as an ellipse, is

ax² + bxy + cy² + dx + ey+ f = 0.

To determine the equation of the comet’s orbit, astronomers can find the coordinates of the comet at five different points (x_i, y_i) where i = 1, 2, 3, 4, and 5, substitute these coordinates into the equation

3.4 Linear Algebra Applied

Elementary Linear Algebra: Section 3.4, p.141

and then expand by cofactors in the first row to find a, b, c, d, e, and f. For example, the coefficient of x² is

Knowing the equation of 67P’s orbit helped astronomers determine the ideal time to release the probe.

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