MATH 166 Spring 2016
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5.1
Chapter 5 Matrices
5.1 Introduction to Matrices
Definition: A matrix A, also denoted by (aij )m×n or simply (aij ), is a rectangular array of numbers.
The order of A is denoted by m × n, where m is the number of rows and n is the number of columns.
The element in the ith row and jth column is denoted by aij .
Example: Find a14 and a43 if



A=

6 −2 5 −7
−17 2 9 3
8
0 4 12
7
1 5 −9





Definition: Two matrices are equal if they have the same order and all corresponding entries are
equal.
Fundamental Operations on Matrices:
• Multiplication of a Matrix by a Number: If c is a number and A is a matrix, then the matrix
cA is the matrix obtained by multiplying every entry in A by c.
• Addition and Subtraction of Matrices: Two matrices of the same order can be added (or subtracted) to obtain another matrix of the same order by adding (or subtracting) corresponding
entries.
• The Transpose of a Matrix: If A is an m × n matrix with elements aij , then the transpose of
A is the n × m matrix AT with elements aji .
Examples:
1. Find the values of a, b, c, and d in the matrix equation below.
a b
c d
+6
7 −6
5 2
T
=
−5 −5
−5 3
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MATH 166 Spring 2016
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Liu
2. A firm makes bulldozers (B), cranes (C) and tractors (T) at two locations, New York City
(NYC) and Los Angeles (LA). The matrices below show the number of each item made in each
location for the months of January (J) and February (F). Suppose the production for March
of all products at all locations was the average of the January and February production. Write
an expression using the matrices J and F for the March production matrix M.
3. The Campus Bookstore’s inventory of books is as follows.
Hardcover: textbooks, 5003; fiction, 1755; nonfiction, 2465; reference, 1833
Paperback: fiction, 2925; nonfiction, 1451; reference, 2038; textbooks, 1619
The College Bookstore’s inventory of books is as follows.
Hardcover: textbooks, 6256; fiction, 2319; nonfiction, 1800; reference, 1583
Paperback: fiction, 3034; nonfiction, 1769; reference, 2807; textbooks, 2186
(a) Represent the Campus’s inventory as a matrix A and the College’s inventory as a matrix
B.
(b) The two companies decide to merge, so now write a matrix C that represents the total
inventory of the newly amalgamated company.
4. Matrix L is a 4 × 6 matrix, matrix M is a 6 × 6 matrix, matrix N is a 4 × 4 matrix, and matrix
P is a 6 × 4 matrix. Find the dimensions of the sums below, if they exist. (If an answer does
not exist, enter DNE.)
(a) L + M
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MATH 166 Spring 2016
5.1
c
Wen
Liu
(b) L + P T
(c) M + N
(d) N + N
Technology Corner: The matrix operations can be carried out by MATRIX command.
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