We proceed the same way to obtain the second row of
In other words, row 2 of
times column 1 of
row 2 of
times column 2 of
row 2 of
times column 3 of
When complete, the product matrix will be
Properties of matrix multiplication
For the matrices
and
the following properties hold.
Matrix multiplication is associative:
Matrix multiplication is distributive:
Note that matrix multiplication is not commutative.
Multiplying two matrices
Multiply matrix
and matrix
First, we check the dimensions of the matrices. Matrix
has dimensions
and matrix
has dimensions
The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions
As the dimensions of
are
and the dimensions of
are
these matrices can be multiplied together because the number of columns in
matches the number of rows in
The resulting product will be a
matrix, the number of rows in
by the number of columns in
The dimensions of
are
and the dimensions of
are
The inner dimensions match so the product is defined and will be a
matrix.
Yes, consider a matrix A with dimension
and matrix B with dimension
For the product AB the inner dimensions are 4 and the product is defined, but for the product BA the inner dimensions are 2 and 3 so the product is undefined.
Using matrices in real-world problems
Let’s return to the problem presented at the opening of this section. We have
[link] , representing the equipment needs of two soccer teams.
Wildcats
Mud Cats
Goals
6
10
Balls
30
24
Jerseys
14
20
We are also given the prices of the equipment, as shown in
[link] .
Goal
$300
Ball
$10
Jersey
$30
We will convert the data to matrices. Thus, the equipment need matrix is written as
The cost matrix is written as
We perform matrix multiplication to obtain costs for the equipment.
The total cost for equipment for the Wildcats is $2,520, and the total cost for equipment for the Mud Cats is $3,840.
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,