2.4
Matrices
1
Section 2.4
Matrices
A matrix is an ordered rectangular array of numbers. A matrix
with m rows and n columns has size or dimension m x n. The
entry in the i th row and j th column is denoted by aij .
A matrix with only one column or one row is called a column
matrix (or column vector) or row matrix (or row vector),
respectively.
The real numbers that make up the matrix are called entries or
elements of the matrix.
7
7 
 2


3
9 
− 5
Example 1: Given A = 
,
0 − 10 20 


1
−
3
11
−


a. What is the dimension of A?
b. Identify a43 .
c. Identify
.
A square matrix is a matrix having the same number of rows as
columns.
 3 9

Example: 
 4 1
Equality of Matrices:
Two matrices are equal if they have the same dimension and their
corresponding entries are equal.
Example 2: Solve the following matrix equation for w, x, y, and z.
 w + 6 x  − 2 0
 y − 2 z  =  1 4

 

2.4
Matrices
2
Addition and Subtraction of Matrices
If A and B are two matrices of the same dimension,
1. The sum A + B is the matrix obtained by adding the
corresponding entries in the two matrices.
2. The difference A – B is the matrix obtained by subtracting the
corresponding entries in B from A.
Laws for Matrix Addition:
If A, B, and C are matrices of the same dimension, then
1. A + B = B + A
2. (A + B) + C = A + (B + C)
Example 3: Refer to the following matrices:
8 − 3 1 
A = 0 − 9 − 4  , B =
 9 6
7 
− 5 4 − 1
 8
, C =
4
8


 10 15 − 2 
 4 1 3
D=

8 5 1
If possible,
a. compute A − B
b. compute B + C.
10 − 8 3
 5 − 4 2 ,


2.4
Matrices
10 − 8 3
C=

 5 − 4 2
3
 4 1 3
D=

8 5 1
c. compute D + C.
Transpose of a Matrix
If A is an m x n matrix with elements aij , then the transpose of A
is the n x m matrix A T with elements a ji .
5 50 
2

A=1
3 27 
16 45 1 
A
T
1 16 
2
5
3 45
=
50 27 1 
Example 4: Given the following matrices, find their transpose.
0
6
− 3
a. B = 

10
100
3


2.4
Matrices
4
 0 


 − 4
b. D = 
11 


 − 3
Scalar Multiplication
A matrix A may be multiplied by a real number, called a scalar in
the context of matrix algebra.
Scalar Product
If A is a matrix and c is a real number, then the scalar product cA
is the matrix obtained by multiplying each entry of A by c.
The zero matrix is one in which all entries are zero.
Note: O represents the zero matrix
2 3
1 2
 − 1 4
 1
 , B = 
 , and C = 
 find,
Example 5: Let A = 
3 4
 − 7 9
 − 6 − 9 1
if possible,
a. -2B – A
b. B + C
Example 6: Solve for the variables in the matrix equation for a
and m.
a + 2 3z + 1 5m  3a 2z 5m 10 − 14 80 
+
=
 4k
0
3  2k 5
6  10
5
9 

2.4
Matrices
5