(Section 8.1: Matrices and Determinants) 8.01
CHAPTER 8: MATRICES and DETERMINANTS
The material in this chapter will be covered in your Linear Algebra class (Math 254 at Mesa).
SECTION 8.1: MATRICES and SYSTEMS OF EQUATIONS
PART A: MATRICES
A matrix is basically an organized box (or “array”) of numbers (or other expressions).
In this chapter, we will typically assume that our matrices contain only numbers.
Example
Here is a matrix of size 2 3 (“2 by 3”), because it has 2 rows and 3 columns:
1 0 2
0 1 5
The matrix consists of 6 entries or elements.
In general, an m n matrix has m rows and n columns and has mn entries.
Example
Here is a matrix of size 2 2 (an order 2 square matrix):
4 1
3 2
The boldfaced entries lie on the main diagonal of the matrix.
(The other diagonal is the skew diagonal.)
, (Section 8.1: Matrices and Determinants) 8.02
PART B: THE AUGMENTED MATRIX FOR A SYSTEM OF LINEAR EQUATIONS
Example
3x + 2 y + z = 0
Write the augmented matrix for the system:
2x z = 3
Solution
Preliminaries:
Make sure that the equations are in (what we refer to now as)
standard form, meaning that …
• All of the variable terms are on the left side (with x, y, and z
ordered alphabetically), and
• There is only one constant term, and it is on the right side.
Line up like terms vertically.
Here, we will rewrite the system as follows:
3x + 2 y + z = 0
2x z=3
(Optional) Insert “1”s and “0”s to clarify coefficients.
3x + 2 y + 1z = 0
2x + 0 y 1z = 3
Warning: Although this step is not necessary, people often
mistake the coefficients on the z terms for “0”s.
CHAPTER 8: MATRICES and DETERMINANTS
The material in this chapter will be covered in your Linear Algebra class (Math 254 at Mesa).
SECTION 8.1: MATRICES and SYSTEMS OF EQUATIONS
PART A: MATRICES
A matrix is basically an organized box (or “array”) of numbers (or other expressions).
In this chapter, we will typically assume that our matrices contain only numbers.
Example
Here is a matrix of size 2 3 (“2 by 3”), because it has 2 rows and 3 columns:
1 0 2
0 1 5
The matrix consists of 6 entries or elements.
In general, an m n matrix has m rows and n columns and has mn entries.
Example
Here is a matrix of size 2 2 (an order 2 square matrix):
4 1
3 2
The boldfaced entries lie on the main diagonal of the matrix.
(The other diagonal is the skew diagonal.)
, (Section 8.1: Matrices and Determinants) 8.02
PART B: THE AUGMENTED MATRIX FOR A SYSTEM OF LINEAR EQUATIONS
Example
3x + 2 y + z = 0
Write the augmented matrix for the system:
2x z = 3
Solution
Preliminaries:
Make sure that the equations are in (what we refer to now as)
standard form, meaning that …
• All of the variable terms are on the left side (with x, y, and z
ordered alphabetically), and
• There is only one constant term, and it is on the right side.
Line up like terms vertically.
Here, we will rewrite the system as follows:
3x + 2 y + z = 0
2x z=3
(Optional) Insert “1”s and “0”s to clarify coefficients.
3x + 2 y + 1z = 0
2x + 0 y 1z = 3
Warning: Although this step is not necessary, people often
mistake the coefficients on the z terms for “0”s.
