Math 1390 - Manyo
4.2 - page 1 of 3
4.2 – Systems of Linear Equations and Augmented Matrices
Read pages 188 - 197
Homework: page196


Matrices: 1, 3, 5, 7, 9, 11, 13, 15, 21, 25, 27
Solving Linear Systems Using Augmented Matrices: 35, 41, 45, 47, 49, 53, 59, 61, 63
Q1: Use this matrix to learn the terminology and answer the questions
1
A
2
3
4
5
6
1. If a matrix has m rows and n columns, it is called an m  n matrix. The numbers m and n are called
the dimensions of the matrix. What are the dimensions of A ? _____________________
2. Each number in a matrix is called an element of the matrix. The position of each element, a i j , is given
using double subscript notation by its row number, i , and its column number, j .
 Using the matrix A, what are a 2 1 = ______ a1 3 = _______
 Rewrite (above) the matrix A using the a i j
3. A matrix with the same number of rows as columns (m = n), is called a square matrix of order n.
4. A matrix with one column is called a column matrix and has dimension m x 1
5. A matrix with one row is called a row matrix and has dimension 1 x n
 Classify each matrix and give its dimension
B  1
2
3
__________________________
3
C  2
3
5 
4
D 
3 
 
2
4 5
4 8 
6 9 
__________________________
_______________________
6. The elements a11, a22 , a33 ,.... make up the elements in the principal diagonal of a matrix.
 List the elements in the principal diagonal of the matrices A and C .
7. The augmented matrix of the system of linear equations
a11x1  a12 x2  k1
a21x1  a22 x2  k 2
 a11 a12 k1 

a 21 a 22 k 2 
is 
Math 1390 - Manyo
4.2 - page 2 of 3
Q2: Set up the augmented matrix for the system
x1  3x2  2
4 x1  6 x2  8
An augmented matrix is transformed into a row – equivalent matrix by performing any of the following
row operations:
 Two rows are interchanged
R
i
 Rj :
Ri and R j are interchanged
 A row is replaced by a nonzero, constant multiple of itself cRi  Ri  : cRi replaces Ri
 A constant multiple of one row is added to another row
cR
j
 Ri  Ri  : cR j  Ri replaces Ri
Q2: Perform the following operations on the augmented matrix from the previous page.
R1  R2
1
  R2  R2
4
(4) R1  R2  R2
1  3 2 


4  6  8
1  3 2 


4  6  8
1  3 2 


4  6  8
Q3: Solve each augmented matrix
1 0 2 

0 3 - 6 
A. 
1 2 4

0 1 1 
B. 
Math 1390 - Manyo
4.2 - page 3 of 3
Q4: In the left column, solve the systems using augmented matrices. To the right is a summary of the possible
final matrix forms for a linear system of two Equations in two variables. Note: m, n and p are real numbers
and p  0
A. −3𝑥1 − 𝑥2 = 5
𝑥1 − 3𝑥2 = −5
Form 1:
A unique solution
Consistent and Independent
1 0 m


0 1 n 
𝑥1 = 𝑚
𝑥2 = 𝑛
B.
𝑥1 − 2𝑥2 = −3
−2𝑥1 + 4 𝑥2 = 6
Form 2:
Infinitely many solutions
Consistent and Dependent
1 m n 


0 0 0 
𝑥1 = 𝑛 − 𝑚𝑘
𝑥2 = 𝑘
C.
𝑥1 − 2𝑥2 = −3
−2𝑥1 + 4 𝑥2 = 8
Form 3:
No solution
Inconsistent
1 m n 


0 0 p 