College Algebra 9th txtbk

458 CHAPTER 7 SYSTEMS OF EQUATIONS AND MATRICES
MATCHED PROBLEM 1
Add:
(A)
3 2 2 3
£ 1 1 § £ 1 1 §
0 3 2 2
(B)
[1 2 7] [2 4 3 1]
Technology Connections
Graphing calculators can be used to solve problems involving
matrix operations. Figure 1 illustrates the solutions to
Example 1A and 1B on a graphing calculator.
(a) Example 1A
(b) Example 1B
Z Figure 1 Matrix addition on a graphing calculator.
0 0 0 0
£ 0 0 0 0§
0 0 0 0
0
0
≥ ¥
0
0
c 0 0
0 0 d
[0 0 0]
Z Figure 2 Zero matrices.
Because we add two matrices by adding their corresponding elements (which are real
numbers), it follows from the properties of real numbers that matrices of the same size are
commutative and associative relative to addition. That is, if A, B, and C are matrices of the
same size, then
A B B A
(A B) C A (B C) Associative
Commutative
A matrix with elements that are all 0’s is called a zero matrix. Examples of zero matrices
are shown in Figure 2.
[Note: “0” is often used to denote the zero matrix of any size.]
The negative of a matrix M, denoted by M, is a matrix with elements that are the
negatives of the elements in M. So if
then
M c a
c
M c a
c
b
d d
Based on our definition of addition, M (M) 0 (a zero matrix).
If A and B are matrices of the same size, then we define subtraction as follows.
A B A (B)
To subtract matrix B from matrix A, we subtract corresponding elements.
b
d d
EXAMPLE 2 Matrix Subtraction
Subtract: c 3 2
5 0 d c 2 2
3 4 d