MATRICES and DETERMINANTS

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
Write the augmented matrix for the system:
3x + 2y + z = 0
?? 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 + 2y + z = 0
?? 2x ?? z = 3
??????
(Optional) Insert “1”s and “0”s to clarify coefficients.
3x + 2y + 1z = 0
?? 2x + 0y ?? 1z = 3
??????
Warning: Although this step is not necessary, people often
mistake the coefficients on the z terms for “0”s.
(Section 8.1: Matrices and Determinants) 8.03
Write the augmented matrix:
Coefficients of Right
x y z sides
3 2 1
??2 0 ??1
0
3
??
?? ??
??
?? ??
Coefficient matrix Right-hand
side (RHS)
Augmented matrix
??????????????????????????????????
We may refer to the first three columns as the x-column, the
y-column, and the z-column of the coefficient matrix.
Warning: If you do not insert “1”s and “0”s, you may want to read the
equations and fill out the matrix row by row in order to minimize the
chance of errors. Otherwise, it may be faster to fill it out column by
column.
The augmented matrix is an efficient representation of a system of
linear equations, although the names of the variables are hidden.
(Section 8.1: Matrices and Determinants) 8.04
PART C: ELEMENTARY ROW OPERATIONS (EROs)
Recall from Algebra I that equivalent equations have the same solution set.
Example
Solve: 2x ?? 1 = 5
2x ?? 1 = 5
2x = 6
x = 3 ?? Solution set is {3}.
To solve the first equation, we write a sequence of equivalent equations until
we arrive at an equation whose solution set is obvious.
The steps of adding 1 to both sides of the first equation and of dividing both
sides of the second equation by 2 are like “legal chess moves” that allowed
us to maintain equivalence (i.e., to preserve the solution set).
Similarly, equivalent systems have the same solution set.
Elementary Row Operations (EROs) represent the legal moves that allow us to write a
sequence of row-equivalent matrices (corresponding to equivalent systems) until we
obtain one whose corresponding solution set is