Math 141 Lecture Notes for Section 2.4
Section 2.2 -
1
Matrices
Previously, we discussed matrices as a tool to be used in finding solutions to systems of equations. Now
we are going to turn our attention to matrices as a tool unto themselves.
Definition 2.2.1:
A matrix is an ordered rectangular array of numbers. A matrix with m rows and n columns is called a
m × n matrix. If we call a matrix M , then the entry in the ith row and the jth column of M is called
mij .
Example 2.2.2:
Consider the following matrix
A=
(i) What are the dimensions of A?
(ii) What is the a22 entry of A?
(iii) What is the a13 entry of A?
2
1
3
5
4
8
Math 141 Lecture Notes for Section 2.4
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There are two special types of matrices we should take note of.
(1) Column matrices are matrices of size m × 1.
(2) row matrices are matrices of size 1 × n.
Lastly, two matrices are equal of they have the same dimensions and equal corresponding entries.
Example 2.2.3:
Given the following matrix equation, what are the values of r, s, t?
1
3r
r−t
6
=
−2 s − 4t
−2 17
Math 141 Lecture Notes for Section 2.4
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We can also add matrices. If we have two matrices that have the same dimensions, we can find their
sum by adding the corresponding entries.
Example 2.2.4:
Compute the following sum:

1
 3
6
2
5
2
 
4
2
8 + 4
7
9
6
7
0

3
5 =
1
Just like with addition, we can also subtract two matrices of the same dimensions. We compute the
difference of their corresponding entries.
Example 2.2.5:
Compute the following difference:

2
 3
7
0
1
8
 
5
4
5 − 2
3
9
2
0
1

6
4 =
7
Math 141 Lecture Notes for Section 2.4
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Another operation commonly performed on matrices is that of transposition. The transpose of a m×n
matrix M is the n × m matrix M T with entries mji .
Example 2.2.6:
Find the transpose M T of the matrix
M=
1
2
3
4
5
6
Math 141 Lecture Notes for Section 2.4
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Lastly, we can compute the scalar multiple of a matrix. Given a matrix M and a number c, we can
compute the scalar multiple of M by c, noted as cM by multiplying each entry of M by the number c.
Example 2.2.7:
Find 3M for the matrix

1
M = 2
5
7
6
3

9
4 
8
Math 141 Lecture Notes for Section 2.4
Example 2.2.8:
Consider the following 3 matrices,


2 7
A= 6 3 
1 4
6
B=
2
4
3
5
1
3
C=
Compute the following:
(i) Find A + C.
(ii) Find B − C.
(iii) Find AT .
Suggested Homework Problems: 3,7,9,11,15,19,21,23,31,33,39
6
5
2
7
3
1