9.5 Matrices and matrix operations  (Page 3/10)

 Page 3 / 10
Lab A Lab B
Computers 15 27
Computer Tables 16 34
Chairs 16 34

Converting the data to a matrix, we have

${C}_{2013}=\left[\begin{array}{c}15\\ 16\\ 16\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}27\\ 34\\ 34\end{array}\right]$

To calculate how much computer equipment will be needed, we multiply all entries in matrix $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ by 0.15.

$\left(0.15\right){C}_{2013}=\left[\begin{array}{c}\left(0.15\right)15\\ \left(0.15\right)16\\ \left(0.15\right)16\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\left(0.15\right)27\\ \left(0.15\right)34\\ \left(0.15\right)34\end{array}\right]=\left[\begin{array}{c}2.25\\ 2.4\\ 2.4\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}4.05\\ 5.1\\ 5.1\end{array}\right]$

We must round up to the next integer, so the amount of new equipment needed is

$\left[\begin{array}{c}3\\ 3\\ 3\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}5\\ 6\\ 6\end{array}\right]$

Adding the two matrices as shown below, we see the new inventory amounts.

$\left[\begin{array}{c}15\\ 16\\ 16\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}27\\ 34\\ 34\end{array}\right]+\left[\begin{array}{c}3\\ 3\\ 3\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}5\\ 6\\ 6\end{array}\right]=\left[\begin{array}{c}18\\ 19\\ 19\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}32\\ 40\\ 40\end{array}\right]$

This means

${C}_{2014}=\left[\begin{array}{c}18\\ 19\\ 19\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}32\\ 40\\ 40\end{array}\right]$

Thus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables, and 40 chairs.

Scalar multiplication

Scalar multiplication involves finding the product of a constant by each entry in the matrix. Given

$A=\left[\begin{array}{cccc}{a}_{11}& & & {a}_{12}\\ {a}_{21}& & & {a}_{22}\end{array}\right]$

the scalar multiple $\text{\hspace{0.17em}}cA\text{\hspace{0.17em}}$ is

Scalar multiplication is distributive. For the matrices $\text{\hspace{0.17em}}A,B,$ and $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ with scalars $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b,$

$\begin{array}{l}\\ \begin{array}{c}a\left(A+B\right)=aA+aB\\ \left(a+b\right)A=aA+bA\end{array}\end{array}$

Multiplying the matrix by a scalar

Multiply matrix $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by the scalar 3.

$A=\left[\begin{array}{cc}8& 1\\ 5& 4\end{array}\right]$

Multiply each entry in $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by the scalar 3.

Given matrix $\text{\hspace{0.17em}}B,\text{}$ find $\text{\hspace{0.17em}}-2B\text{\hspace{0.17em}}$ where

$B=\left[\begin{array}{cc}4& 1\\ 3& 2\end{array}\right]$

$-2B=\left[\begin{array}{cc}-8& -2\\ -6& -4\end{array}\right]$

Finding the sum of scalar multiples

Find the sum $\text{\hspace{0.17em}}3A+2B.$

First, find $\text{\hspace{0.17em}}3A,\text{}$ then $\text{\hspace{0.17em}}2B.$

$\begin{array}{l}\begin{array}{l}\hfill \\ \hfill \\ 3A=\left[\begin{array}{lll}3\cdot 1\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}3\left(-2\right)\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}3\cdot 0\hfill \\ 3\cdot 0\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}3\left(-1\right)\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}3\cdot 2\hfill \\ 3\cdot 4\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}3\cdot 3\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}3\left(-6\right)\hfill \end{array}\right]\hfill \end{array}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left[\begin{array}{rrr}\hfill 3& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-6& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ \hfill 0& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-3& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}6\\ \hfill 12& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}9& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-18\end{array}\right]\hfill \end{array}$
$\begin{array}{l}\begin{array}{l}\hfill \\ \hfill \\ 2B=\left[\begin{array}{lll}2\left(-1\right)\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}2\cdot 2\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}2\cdot 1\hfill \\ 2\cdot 0\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}2\left(-3\right)\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}2\cdot 2\hfill \\ 2\cdot 0\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}2\cdot 1\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}2\left(-4\right)\hfill \end{array}\right]\hfill \end{array}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left[\begin{array}{rrr}\hfill -2& \hfill 4& \hfill 2\\ \hfill 0& \hfill -6& \hfill 4\\ \hfill 0& \hfill 2& \hfill -8\end{array}\right]\hfill \end{array}$

Now, add $\text{\hspace{0.17em}}3A+2B.$

Finding the product of two matrices

In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is an matrix and $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ is an matrix, then the product matrix $\text{\hspace{0.17em}}AB\text{\hspace{0.17em}}$ is an matrix. For example, the product $\text{\hspace{0.17em}}AB\text{\hspace{0.17em}}$ is possible because the number of columns in $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is the same as the number of rows in $\text{\hspace{0.17em}}B.\text{\hspace{0.17em}}$ If the inner dimensions do not match, the product is not defined.

We multiply entries of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ with entries of $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.

To obtain the entries in row $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ of $\text{\hspace{0.17em}}AB,\text{}$ we multiply the entries in row $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by column $\text{\hspace{0.17em}}j\text{\hspace{0.17em}}$ in $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ and add. For example, given matrices $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B,\text{}$ where the dimensions of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ are and the dimensions of $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ are the product of $\text{\hspace{0.17em}}AB\text{\hspace{0.17em}}$ will be a matrix.

Multiply and add as follows to obtain the first entry of the product matrix $\text{\hspace{0.17em}}AB.$

1. To obtain the entry in row 1, column 1 of $\text{\hspace{0.17em}}AB,\text{}$ multiply the first row in $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by the first column in $\text{\hspace{0.17em}}B,$ and add.
$\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right]\cdot \left[\begin{array}{c}{b}_{11}\\ {b}_{21}\\ {b}_{31}\end{array}\right]={a}_{11}\cdot {b}_{11}+{a}_{12}\cdot {b}_{21}+{a}_{13}\cdot {b}_{31}$
2. To obtain the entry in row 1, column 2 of $\text{\hspace{0.17em}}AB,\text{}$ multiply the first row of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by the second column in $\text{\hspace{0.17em}}B,$ and add.
$\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right]\cdot \left[\begin{array}{c}{b}_{12}\\ {b}_{22}\\ {b}_{32}\end{array}\right]={a}_{11}\cdot {b}_{12}+{a}_{12}\cdot {b}_{22}+{a}_{13}\cdot {b}_{32}$
3. To obtain the entry in row 1, column 3 of $\text{\hspace{0.17em}}AB,\text{}$ multiply the first row of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by the third column in $\text{\hspace{0.17em}}B,$ and add.
$\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right]\cdot \left[\begin{array}{c}{b}_{13}\\ {b}_{23}\\ {b}_{33}\end{array}\right]={a}_{11}\cdot {b}_{13}+{a}_{12}\cdot {b}_{23}+{a}_{13}\cdot {b}_{33}$

The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations