Lesson 1: Matrices and Vectors
When you think of arithmetic, you probably think of adding, subtracting, multiplying and di
viding
numbers
. However, basic arithmetic in Matlab is concerned with
matrices
. In this lesson, we
recall what matrices are and what it means to perform various arithmetic operations with them.
This will lay the groundwork for introducing Matlab in the next lesson.
General Terminology
A
matrix
is a rectangular array of numbers. It is usually written with brackets enclosing the
numbers on either side. The following are all examples of matrices.
a)
A
=
1
4
1
1
−
1
0
6
5
b)
B
=
2
−
1
0
3
5
4
0
3
−
7
c)
C
=
3
.
2
−
2
.
5
4
.
0
10
.
7
−
5
.
4
d)
D
=
0
1
2
e)
E
=
17
The number of rows and columns is called the
dimension
of the matrix.
For instance, the
matrix
A
above has 2 rows and 4 columns and is referred to as a 2
×
4 (read, “two by four”) matrix.
The matrix
B
is 3
×
3,
C
is 1
×
5,
D
is 3
×
1, and
E
is 1
×
1. Notice that the number of rows is
always stated first followed by the number of columns. Thus,
D
is a 3
×
1 matrix and not a 1
×
3
matrix. If a matrix has the same number of rows as columns then it is called a
square matrix
.
The matrices
B
and
E
above are both square matrices. If a matrix has only one row or only one
column then it can also be called a
vector
. More specifically, a matrix that has only one row is
called a
row vector
and a matrix with only one column is called a
column vector
. For instance,
C
,
D
, and
E
are all vectors;
C
is a row vector,
D
is a column vector, and
E
is both a row vector
and a column vector. Notice that a 1
×
1 matrix such as
E
above is simply a number. It can be
written without brackets surrounding it. It is both a row vector and a column vector, but is most
commonly referred to as a
scalar
.
The rows of a matrix are numbered from top to bottom and the columns are numbered from left
to right. For example, the first row of
A
is
1
4
1
1
and the second column is
4
0
.
The numbers in a matrix are called
entries
. They can be identified by the row and column that
they are in. For example, the entry in the second row and first column of
A
is 1 and the entry in
the second row and third column of
B
is 4. If
M
is a matrix, then
M
ij
refers to the entry in its
i
’th
row and
j
’th column. For example
A
21
=
−
1 and
B
23
= 4. Notice that, just as the dimension of a
matrix is the number of rows followed by the number of columns, an entry in a matrix is specified
by stating first its row number and then its column number. Thus,
A
21
=
−
1 but
A
12
= 4. Two
matrices are
equal
if and only if they have the same dimension and their corresponding entries are
equal to each other.
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The
transpose
of a matrix
M
is the matrix that is obtained when the rows of
M
are turned into
columns (and vice versa). The transpose of
M
is denoted
M
t
. For instance, consider the matrix
A
above. To form
A
t
we take the first row of
A
1
4
1
1
and write it as a column,
1
4
1
1
.
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 '09
 WETZER
 Multiplication, Massachusetts

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