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Curl of a Vector Field
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Definition of a Matrix
Current Location
>
Math Formulas
>
Linear Algebra
> Matrix Multiplication
Matrix Multiplication
We can only multiply two
matrices
if their
dimensions
are
compatible
, which means the number of columns in the first matrix is the same as the number of rows in the second matrix.
If
A
= [a
_{i}
_{ j}
] is an
m
×
n
matrix and
B
= [b
_{i}
_{ j}
] is an
n
×
p
matrix, the product
AB
is an
m
×
p
matrix.
AB
= [c
_{i}
_{ j}
], where
c
_{i}
_{ j}
=
a
_{i}
_{1}
b
_{1 j }
+
a
_{i}
_{2}
b
_{2 j}
+ … +
a
_{in}
b
_{n j}
.
(The entry in the
i
^{th}
row and
j
^{th}
column is denoted by the double subscript notation
a
_{i}
_{ j}
,
b
_{i}
_{ j}
, and
c
_{i}
_{ j}
. For instance, the entry
a
_{23}
is the entry in the second row and third column.)
The definition of matrix multiplication indicates a rowbycolumn multiplication, where the entries in the
i
^{th}
row of
A
are multiplied by the corresponding entries in the
j
^{th}
column of
B
and then adding the results.
Matrix multiplication is NOT commutative. If neither A nor B is an identity matrix,
AB
≠
BA
.
How to multiply a Row by a Column?
We'll start by showing how to multiply a 1
× n
matrix by an
n
× 1 matrix. The first is just a single row, and the second is a single column. By the rule above, the product is a 1 × 1 matrix; in other words, a single number.
First, let's name the entries in the row
r
_{1}
,
r
_{2}
, ...,
r
_{n}
, and the entries in the column
c
_{1}
,
c
_{2}
, ...,
c
_{n}
. Then the product of the row and the column is the 1 × 1 matrix
[
r
_{1}
c
_{1}
+
r
_{2}
c
_{2}
+
^{...}
+
r
_{n}
c
_{n}
].
Example 1:
Multiply
Solution:
Here, the number of columns in the first matrix is the same as the number of rows in the second matrix. Hence the dimension of the resultant matrix would be 2 × 2.
To start with, we'll concentrate on working out the first element of the resultant matrix, element a
_{11}
. (That's the element in the first row and the first column of the resultant matrix.) The first thing to do is to multiply the 0 in the first matrix by the 3 in the second matrix. Then, moving across the top row in the first matrix and down the first column in the second matrix, we multiply the 1 by the 1.Then, moving across the top row in the first matrix and down the first column in the second matrix, we multiply 2 by 6.Finally we add the results of those three multiplications to get the element a
_{11}
in the result matrix.
That gives 0 × 3 + (– 1×1) + 2 × 6 = 0 – 1 + 12 = 11. So the element a
_{11}
in the result matrix is 11.
In the same fashion, we get:
a
_{12}
= 0 × (– 1) + (– 1 × 2) + 2 ×1 = – 2 + 2 = 0
a
_{21}
= 4 × 3 + 11 × 1 + 2 × 6 = 12 + 11 + 12 = 35
a
_{22}
= 4 × (–1) + 11 × 2 + 2 × 1 = –4 + 22 + 2 = 20
Hence the resultant matrix is
Example 2:
Multiply these matrices:
Solution:
Here, the dimension of both the matrices is same, so the resultant matrix will also have the same dimension.
Example 3:
What is the resultant of
?
Solution:
Here, the number of columns in the first matrix is the same as the number of rows in the second matrix. Hence the dimension of the resultant matrix would be 2 × 1.
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