TERMINOLOGY - BASIC THEOREMS, EXAMPLES & EXERCISES
Linear Systems and Matrices.
Linear equations are equations that involve only the first powers of the variables.
Exercise 1: Classify the following equations as linear or nonlinear:
x  y  y 2  10, xy  x  y  12  0, x 2  y 3  1, x  2 y  6 z  7, x  y  2  0
A system of linear equations (also called a linear system) is a finite collection of linear
equations.
Example 1:
x  2 y  6 z  7,
x  y  2 z  10
x  2 y  3z  4
Note: If at least one of the constant terms in the linear system is nonzero, the linear
system is called nonhomogeneous, otherwise, the linear system is called homogeneous.
By a solution of a system is meant a set of values (normally real numbers) for the
variables that satisfy all the equations in the system simultaneously.
Example 2: The set of values x = 23, y = -11, and z = 1 constitute a solution of the
system
x  2 y  6 z  7,
x  y  2 z  10
x  2 y  3z  4
because
23  2(11)  6(1)  7,
23  (11)  2(1)  10
23  2(11)  3(1)  4
A linear system is said to be consistent if it has at least one solution
Example 3: The system in Example 2. is consistent.
Exercise 2: Show that the following system is inconsistent.
2 y  6 z  7,
y  3 z  10
Exercise 3: Give the geometrical interpretations for each of the following:
The system
a1 y  b1 z  c1 ,
a 2 y  b2 z  c 2
(a) has a unique solution;
(b) has an infinite solution set;
(c) has no solution.
Theorem: A linear system of equations either has a unique solution, has no solution, or
has infinitely many solutions.
Note: Every nonhomogeneous linear system with more variables than equations either
has no solution or infinitely many solutions.
Exercise 4: Determine whether the following system has a unique solution, no solution,
or an infinite solution set. If the system has an infinite solution set, express the solution
set in parametric form:
3 x  12 y  9,
2x  8 y  6
Exercise 5: Use the method of elimination to determine if the following linear system is
consistent or inconsistent. If the system is consistent, find the solution if it is unique,
otherwise, describe the infinite solution set in terms of arbitrary parameters.
x  2 y  6 z  7,
x  y  2 z  10
x  2 y  3z  4
Exercise 6: Write the augmented coefficient matrix of the linear system in Exercise 5
The three Elementary Row Operations (ERO) on a matrix A are:
1. Interchange any two rows of matrix A ( Rij );
2. Multiply any one row of A by a nonzero constant ( kRi );
3. Add a constant multiple of one row of A to another row ( kRi  R j ).
Exercise 7: Let
2 1 3 1 
A  3 4 7 6 
1 7 4 6
Perform the following sequence of row operations on the matrix A:
R13 ,  3R1  R2 , - 2R1  R3
A matrix A is called an echelon matrix if it has the following two properties:
1. Any nonzero row of A precedes any zero row (if any);
2. In each nonzero row of A, the first nonzero element lies strictly to the right of the first
nonzero element in the preceding nonzero row (if there is a preceding nonzero row).
Example 4: The following is a list of some 3 x 3 echelon matrices.
2 0 3 1 3  2 1 2 0  1 4 5 
0 0 0  , 0 1 5  ,  0 2 0  ,  0 0 5 

 
 
 

0 0 0 0 0 0  0 0 3  0 0 0
Example 5: The augmented matrix of the system
x  2 y  6 z  2u  v  10,
 3v  2
u  3v  6
z
is the echelon matrix
1 2
0 0

0 0
6
1
0
10
0 3  2 
1 3
6 
2
1
Any matrix can be transformed into an echelon matrix using elementary row operations.
This procedure is known as Gaussian elimination.
Exercise 8: Use EROs to transform the augmented coefficient matrix of the following
system to echelon form. Then solve the system by backward substitution.
3x1  6 x 2  2 x3  1
2 x1  4 x 2  x3  17
x1  2 x 2  2 x3  9
A reduced echelon matrix A is an echelon matrix that has the following two additional
properties:
3. Every leading entry of A is 1;
4. Each leading entry of A is the only nonzero element in its column.
Example 6: The following is a list some reduced echelon matrices.
1 0 3  1 0  2
1 4 0 0
0 0 0, 0 1 5 , 1 0 0, 0 0 1 0

 
 0 1 0 

 0 0 0 1
0 0 0 0 0 0  


The process of transforming a matrix into a reduced echelon form by EROs is called
Gauss-Jordan elimination.
Example 7: Use the Gauss-Jordan elimination to solve the system:
x  2y  z  3
x  3y  z  5
3 x  8 y  4 z  17
Reduce the augmented matrix to a reduced echelon form using the following sequence of
row operations:
1
 R1  R2 ,  3R1  R3 ,  2 R2  R3 , R3 , R3  R1 ,  2 R3  R2 ,  2 R2  R1 .
3
1
1

3
3
8
1
 0
0
2
1
0
2
1
3 1 2  1
1
5   0 1
2
4 17  0 2
7
1
3  1 2
0


2
2   1 1
0
1 4 / 3 3 0
1
3  1 2
2  0 1
8  0 0
13 / 3  1
 2 / 3  0
4 / 3  0
1
2
3
0
1
0
3
2
4 
0
17/ 3 
0  2 / 3
1
4 / 3 
The unique solution of the system is x  17 / 3, y  2 / 3, z  4 / 3
Theorem: Every matrix is row equivalent to one and only one reduced echelon form.
Recall: If the constant terms in the linear system are all zero, then the linear system is
called homogeneous,
Example 8:
x  2 y  3z  0
2x  5 y  2z  0
3x  y  4 z  0
Note:
1. The trivial solution x1  0, x2  0, ..., xn  0 is a solution of every homogeneous
linear system (i.e., a homogeneous system is always consistent);
2. If a linear system has one nontrivial solution, then it must have infinitely many
solutions;
Theorem: Every homogeneous linear system with less equations than variables has
infinitely many solutions.
Theorem: A homogeneous system consisting of n equations in n variables has only the
trivial solution if, and only if, the coefficient matrix of the system is row equivalent to the
n x n identity matrix.
Exercise 10: Show that the homogeneous system in Example 8 has only the trivial
solution.
An n-column vector ( or simply a n-vector) is an n x 1 matrix (i.e. one having n rows
and one column)
Example 9:
1 
 2
a=  
3 
 
 4
is a 4-column vector which is sometimes written as
a = (1, - 2, 3, - 4).
A n-row vector is a 1 x n matrix (i.e. one having one row and n columns)
Example 10:
b = 1  2 3  4 is a 4-row vector.
Matrix Multiplication (Special Case): The product of an n-row vector
b 
 
b2 
. 
A  a1 a2 ... an  and an n-column vector B    is defined as follows:
. 
. 
 
bn 
A  B = a1
b1 
b 
 2
. 
a2 ... an  .   = a1b1  a 2 b2  ...  an bn
. 
. 
 
bn 
Matrix Multiplication (General Case): Suppose A is an m x n matrix and B is an n x p
matrix ( i.e. the number of columns in A is equal to the number of rows in B). Then
the product AB is the m x p matrix whose ij-entry is obtained by multiplying the the ith
row of A by the jth column of B.
Example 11: Let
 1
A
 1
Find (a) AB (b) BA
 1 2  2
(a) AB  

 1 3  3
(b) BA is not defined.
3
4
2
2
and B  

3
3
0  1  2  2  3

6  1  2  3  3
3
4
1 3  2  4
-1 3  3  4
0
6
1  0  2  6 8 11 12

- 1  0  3  6  7 9 18 
Note: Matrix multiplication is not commutative.
Exercise 6: Write the linear system in Exercise 5 in the form AX = B, where A is the
coefficient matrix, X is the column matrix of variables, and B is the column matrix of
constants.
The n x n identity matrix I n is the diagonal matrix with 1s on the main diagonal and 0s
elsewhere.
Example 12: The 3 x 3 identity matrix is
1 0 0
I 3  0 1 0
0 0 1
Note:
If AI n and I n B are defined, then AI n  A and I n B  B
The jth basic unit n-vector is the n x 1 column matrix
0 
. 
 
. 
 
. 
e j  1  1 in the jth row
 
. 
. 
 
. 
0 
 
A square matrix A is called invertible if there exists a matrix B such that
AB = BA = I
A matrix B such that AB = BA = I is called the inverse matrix of the matrix A
Note: The inverse matrix B is denoted by A 1 ;
If the matrix A is invertible then it has a unique inverse.
Theorem: (Inverses of 2 x 2 matrices)
The matrix
a b 
A

c d 
is invertible if and only if ad  bc  0. In this case
 d -b
A 1  

 c a 
The matrix obtained by performing a single ERO on the identity matrix I n is called an
elementary matrix.
A square matrix is called nonsingular if it is invertible.