Math 141 Lecture Notes for Section 2.4
Section 2.4 -
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 2.4.1:
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 2.4.2:
Consider the following matrix
A=
2
1
3
5
4
8
(i) What are the dimensions of A?
A is a 2 × 3 matrix.
(ii) What is the a22 entry of A?
a22 = 5.
(iii) What is the a13 entry of A?
a13 = 4.
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 2.4.3:
Given the following matrix equation, what are the values of r, s, t?
1
3r
r−t
6
=
−2 s − 4t
−2 17
Since each entry of the two matrices must be equal, this gives us the following three equations:
3r = 6
;
r−t=1
and
s − 4t = 17.
The first equation gives us r = 2, the second gives us t = 1 and the 3rd gives us s = 21.
Math 141 Lecture Notes for Section 2.4
2
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 2.4.4:
Compute the following sum:

1
 3
6
 
4
2
8 + 4
7
9
2
5
2
6
7
0

3
5 =
1
Since we add by using entry position, we can rewrite the sum of the two matrices as a single matrix,


1+2 2+6 4+3
 3+4 5+7 8+5 
6+9 2+0 7+1
then we compute all of the individual sums,


3 8 7
 7 12 13 
15 2 8
and we are done.
Just like with addition, we can also subtract two matrices of the same dimensions. We compute the
difference of their corresponding entries.
Example 2.4.5:
Compute the following difference:

2
 3
7
0
1
8
 
4
5
5 − 2
9
3
Just like addition, we can rewrite the difference

2−5
 3−2
7−3
2
0
1

6
4 =
7
of the two matrices as a single matrix,

0−2 4−6
1−0 5−4 
8−1 9−7
then we can compute all of the individual differences to obtain


−3 −2 −2
 1
1
1 
4
7
2
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 .
Math 141 Lecture Notes for Section 2.4
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Example 2.4.6:
Find the transpose M T of the matrix
M=
1
2
3
4
5
6
M is a 2 × 3 matrix so it’s transpose M T must be a 3 × 2 matrix. This matrix is


1 2
MT =  3 4 
5 6
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 2.4.7:
Find 3M for the matrix

1
M = 2
5

9
4 
8
7
6
3
Computing the value 3M written as

1
3M = 3  2
5
we can rewrite it as
7
6
3

9
4 
8
 
3
3·1 3·7 3·9
3M =  3 · 2 3 · 6 3 · 4  =  6
15
3·5 3·3 3·8


21 27
18 12 
9 24
and we are done.
Example 2.4.8:
Consider the following 3 matrices,


2 7
A= 6 3 
1 4
B=
2
4
3
5
1
3
C=
6
5
2
7
3
1
Compute the following:
(i) Find A + C. This does not exist, A is a 3 × 2 matrix and C is a 2 × 3 matrix. Therefore we
cannot compute this sum.
Math 141 Lecture Notes for Section 2.4
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(ii) Find B − C.
2
4
3
5
1
3
−
6
5
2
7
3
1
2−6 3−2 1−3
=
4−5 5−7 3−1
−4
1 −2
.
−1 −2
2
(iii) Find AT .
2
7
6
3
1
4
Suggested Homework Problems: 3,7,9,11,15,19,21,23,31,33,39
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