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Definition of a Matrix
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Linear Algebra
> Definition of a Matrix
Definition of a Matrix
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A rectangular array of numbers is called a matrix. We shall mostly be concerned with matrices having real numbers as entries. The horizontal arrays of a matrix are called its ROWS and the vertical arrays are called its COLUMNS. A matrix having
m
rows and
n
columns is said to have the order
m
×
n
.
A matrix
A
of order
m
×
n
can be represented in the following form:
where
a
_{ij}
is the entry at the intersection of the
i
^{th}
row and
j
^{th}
column.
In a more concise manner, we also denote the matrix
A
by [
a
_{ij}
] by suppressing its order.
Note:
Some books also use
to represent a matrix.
A matrix having only one column is called a column vector and a matrix with only one row is called a row vector. Whenever a vector is used, it should be understood from the context whether it is a row vector or a column vector.
Here are a couple of examples of different types of matrices:
Symmetric:
Diagonal
Upper Triangular
Lower Triangular
Zero
Identity
Example 1:
Let
, list out the
a
_{ij}
’s values in
A
.
Solution
:
a
_{11}
= 1, a
_{12}
= 3, a
_{13}
= 6
a
_{21}
= 2, a
_{22}
= 3, a
_{23}
= 7
a
_{31}
= 4, a
_{32}
= 4, a
_{33}
= 0
Example 2:
State the
a
_{ij}
’s values in
.
Solution:
a
_{11}
= 9, a
_{12}
= 8, a
_{13}
= 7
a
_{21}
= 6, a
_{22}
= 5, a
_{23}
= 4
a
_{31}
= 3, a
_{32}
= 2, a
_{33}
= 1
a
_{41}
= 4, a
_{42}
= 6, a
_{43}
= 8
Example 3:
Provide two examples of column and row matrices each.
Solution
:
We know that, a matrix having only one column is called a column vector or column matrix and a matrix with only one row is called a row vector or row column.
Example 4:
Is the following matrix classified under the category of matrices?
Solution
:
A
is not a matrix because column three or we can say row three is incomplete.
Example 5
:
What is the order of the following matrix?
Solution
:
We know that a matrix having
m
rows and
n
columns is said to have the order
m
×
n,
therefore the order of
A
is 4
×
3.
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