1. Chapter 2
Determinants

2. DETERMINANTS BY COFACTOR EXPANSION
2.1
DETERMINANTS BY COFACTOR EXPANSION
For A (2x2) matrix
The expression is called the determinant of the matrix and is denoted by the symbol det A or |A|
the formula for A-1 given in Theorem is

3. Minors and Cofactors

4. EXAMPLE 1 Finding Minors and Cofactors
Let

6. Cofactor Expansions
The definition of a 3x3 determinant in terms of minors and cofactors is
the determinant of an nxn matrix to be
This method of evaluating det (A) is called cofactor expansion along the first row of A.

7. EXAMPLE 2 Cofactor Expansion Along the First Row
Let .
Evaluate det (A) by cofactor expansion along the first row of A.

8. All of the following are correct for 3x3 A

10. EXAMPLE 3 Cofactor Expansion Along the First Column
Let A be the matrix in Example 2. Evaluate det (A) by cofactor expansion along the first column of .

11. EXAMPLE 4 Smart Choice of Row or Column
If is the 4X4 matrix
then to find det(A) it will be easiest to use cofactor expansion along the second column, since it has the most zeros:
For the determinant, it will be easiest to use cofactor expansion along its second column, since it has the most zeros:

12. Adjoint of a Matrix

13. EXAMPLE 6 Adjoint of a Matrix
Let
The cofactors of A are
so the matrix of cofactors is
and the adjoint of A is

14. THEOREM 2.1.2
Inverse of a Matrix Using Its Adjoint
If A is an invertible matrix, then

15. EXAMPLE 7 Using the Adjoint to Find an Inverse Matrix
Use 7 to find the inverse of the matrix A in Example 6.

16. THEOREM 2.1.3

17. EXAMPLE 8 Determinant of an Upper Triangular Matrix

18. Cramer's Rule

19. EXAMPLE 9 Using Cramer's Rule to Solve a Linear System
Use Cramer's rule to solve

21. Let A be a square matrix. Then
2.2
EVALUATING DETERMINANTS BY ROW REDUCTION
THEOREM 2.2.1
THEOREM 2.2.2
Let A be a square matrix. Then

22. Elementary Row Operations

24. THEOREM 2.2.4

25. EXAMPLE 2 Determinants of Elementary Matrices
The following determinants of elementary matrices, which are evaluated by inspection, illustrate Theorem

26. Matrices with Proportional Rows or Columns
THEOREM 2.2.5
If A is a square matrix with two proportional rows or two proportional columns, then
EXAMPLE 3 Introducing Zero Rows
The following computation illustrates the introduction of a row of zeros when there are two proportional rows:
Each of the following matrices has two proportional rows or columns; thus, each has a determinant of zero.

27. Evaluating Determinants by Row Reduction
EXAMPLE 4 Using Row Reduction to Evaluate a Determinant
Evaluate det(A) where

28. Reduced A to row-echelon form (which is upper triangular) and apply Theorem 2.2.3:

29. EXAMPLE 5 Using Column Operations to Evaluate a Determinant
Compute the determinant of

30. EXAMPLE 6 Row Operations and Cofactor Expansion
Evaluate det (A) where

31. 2.3
PROPERTIES OF THE DETERMINANT FUNCTION
Basic Properties of Determinants
For example,

32. EXAMPLE 1
Consider

33. THEOREM 2.3.1
EXAMPLE 2 Using Theorem 2.3.1

34. THEOREM 2.3.3
A square matrix A is invertible if and only if

35. EXAMPLE 3 Determinant Test for Invertibility
Since the first and third rows of
are proportional, Thus A is not invertible.

36. THEOREM 2.3.4

37. THEOREM 2.3.5
If A is invertible, then

38. Linear Systems of the Form
Many applications of linear algebra are concerned with systems of n linear equations in n unknowns that are expressed in the form
where λ is a scalar
EXAMPLE 5 Finding
The linear system
can be written in matrix form as

39. This is called the characteristic equation of A
EXAMPLE 6 Eigenvalues and Eigenvectors
Find the eigenvalues and corresponding eigenvectors of the matrix A in Example 5.

40. The eigenvectors of A corresponding to λ=-2 are

41. THEOREM 2.3.6

42. 2.4
A COMBINATORIAL APPROACH TO DETERMINANTS
EXAMPLE 7 Determinants of 2x2 and 3x3 Matrices
Warning the methods do not work for determinants of 4x4 matrices or higher.

43. EXAMPLE 8 Evaluating Determinants

44. C H A P T E R 3 Vectors in 2-Space and 3-Space

45. 3.1
INTRODUCTION TO VECTORS (GEOMETRIC)

46. DEFINITION
If v and w are any two vectors, then the sum v + w is the vector determined as follows: Position the vector w so that its initial point coincides with the terminal point of v. The vector v + w is represented by the arrow from the initial point of v to the terminal point of w

49. Vectors in Coordinate Systems

54. Vectors in 3-Space
each point P in 3-space has a triple of numbers (x, y, z), called the coordinates of P

55. In Figure a the point (4, 5, 6) and in Figure b the point (-3 , 2, -4).

57. EXAMPLE 1 Vector Computations with Components

59. EXAMPLE 2 Finding the Components of a Vector

60. Translation of Axes
The solutions to many problems can be simplified by translating the coordinate axes to obtain new axes parallel to the original ones.
These formulas are called the translation equations.

63. EXAMPLE 3 Using the Translation Equations

65. 3.2
NORM OF A VECTOR; VECTOR ARITHMETIC
Properties of Vector Operations
THEOREM 3.2.1
Properties of Vector Arithmetic
If u, v, and w are vectors in 2- or 3-space and k and l are scalars, then the following relationships hold.

67. Norm of a Vector

70. EXAMPLE 1 Finding Norm and Distance