Matrix Properties .First Topic (Shweta & Maria)

Matri ces

Matrix Properties………………………………………….First topic (Shweta & Maria)

Operations………………………………………………..Second topic (Aaron)

Applications……………………………………………….Third topic (Jaun)

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Reference Sheet: ___Matrix Properties___

1. State the term :

Matrix Properties

2. State the mathematical definition of the term.

Shows basinc information that should be acknowledged when working with Matrices.

3. Explain the meaning of the term in your own words.

The Matrix Properties shows basic Vocabulary, how the matrix works, and the main parts of a matrix (such as the determinant) and important things that you need to know and keep in mind.

4. Show an example of the term and show something that looks like an example but is not. Explain what the difference between the two examples is.

Vocabulary:

When working on matrices in a classroom, you will most likely work on them while you are learning about linear sytems.

Matrix: a rectangular or squarearray of numbersthat can be written using brackets.

Dimentions of a matrix: depend on the matrix itself (how many rows and columns it has), first you state the rows and then the columns – example: this matrix has 2 rows (2 &3) and has 2 columns (2&5) so it is a 2x2 matrix.

Matrix Addition: when you add 2 matrices you end up with one. In order to add 2 matrices they have to have the same dimentions. Along with having the same dimetions you have to add them in a special order-example: + = =

Matrix Subtraction: when you subtract 2 matrices you end up with one. In order to subctract the matrices have to have the same dimentions and just like additon they have to be subtracted in a specil order-example: - = =

Matrix Multiplication: As metioned before matrices have a way to be identified by their dimetions; Matrices can only be multiplied if they have compatible dimentions.

Using the A m*n X D n*p “formula” you can identify if the matrices are compatible, since your result will be m*p.

Example:

x = since they are both 2x2 matrices they are compatible with the formula: A m*n X D n*p , where both of the “n’s” have to be equal so 2(n) and 2(n) are out of the “game” and you are left with 2(m) and 2(p) which tells you how your result is going to be which in this case is a 2 by 2 (2x2).

*when you actually multiply you have to know that in order to do that you multiply rows by columns as shown below:

x First you multiply your first row (a c) by the columns (w x) and (y z ), you start with: : a*w, then you multiply c*x and add them up to find your top left spot.

Then you multiply a*y and c*z and add them to get your top right spot.

After you have done that you move on to the second row (b d) and start multiplying it as the first one: b*w, the d*x and add them to get your bottom left spot. Move on to b*y and d*z and add them to get your right bottom spot.

Your matrix will end up looking like this:

Determinant of a matrix: used to solve the systems of equations in matrices. The determinant is

found through this formula- Determinant = =(a1b2)- (a2b1)

Equal Matrices: This involves you finding the equivelant of the variable to the corresponding number in the matrix.

Example:

X1= -3 x2= -4 y1= 2 y2= -9

Multiplicative Inverse: Can be done with a Calculator or using a formula

~The Determinant of 2x2 Matrix is ad - bc

~If the Determinant is 0, then there is no inverse.

~ If the Determinant is not equal to 0, then there is an inverse - a and d switch their positions and c and b will change their sign. ( )

Example: Finding Inverse Matrix

Step 1 – Find the Determinant.

(-2)(-4) – (2)(5) = -2 since the Determinant is not 0, then there is an Inverse. Step 2 – Switch -2 and -4’s positions(positions) and change the signs of 2 and 5.

Step 3 – Use the determinant to find Inverse.

= Using the formula

Using the Calculator –

Plug in the original matrix pressing the 2nd button and the Inverse(x-1) of the matrix. Then go to edit and press [A] and then ENTER. Then plug in the dimentions and then the number of the matrix. Press QUIT, select the matrix and under the names tab select the corresponding(the matrix you are working on) matrix, press ENTER and then select the Inverse (x-1), then press ENTER to get the answer.

Identity Matrix: This is when you have a matrix with 1’s in a diagonal line then you have your identity matrix.

Example-

Ex -1 Ex- 2

Non-examples- (commonly made mistakes)

x-y+2z=10

-x+y-2z=5

3x-3y+6z=-2

= 0 Impossible (it is Impossible)

This also applies to the Multiplicative Inverse.

+ = Not possible because they have different dimensions.

Multiplication x = Not possible because the matrices’s dimesions are compatible.

Important things to know:

*When dealing with a matrix you can add(+), subtract(-), and multiply(x), but you can not divide. *When adding and subtracting the matrices have to have the same dimentions.

*When multiplying your matrices have to have compatible dimentions.

When adding or subtracting matrices the dimensions have to be the same. When you have to solve for the matrix, you have to solve them in a corresponding way. Ex- first number on the matrix with the first number on the second matrix.

*The Determinant needs to be a number that is not zero(0), because you cannot divide by zero so it will always end up impossible. However, if the determinant does not equal to zero then you can always find a solution for x, y and z. (

*When workiong with matrices and the associative property of addition A+B = B+A.

5. Show another way that you can think about this term or something that will help you remember the concept. You may want to show a picture, graph, diagram, or drawing that will help you remember the big idea.

*A way to remember that rows come before columns when identifying the dimmentions of the matrix is the soda named RC, since rows come before columns.

How to Write a Skill Description

1. Skill Title (A title gives you a short way to remember what the skill is)

Adding Matrices: combining and adding Subtracting Matrices: combining and subtracting

Multiplying Matrices: twisted combination between rows and columns involving multiplication as well as addition.

2. Skill Description (Few sentances that describes what the skill is and what it is good for. This description should be short enough to read quickly and decide if you can use it to solve the problem you are working on.)

Using matrices helps you have to organize information that is “messy” and where you are being asked to find several specific things.

Using a matrix- you can decide to use a matrix if you have many numbers, names and data.

3. Examples (Use one or two challenging and relevant examples)

Having been in a class together for several weeks now Max, Jenny, Ray and Nancy who are good friends support each other to have goos grades in their classes. They gathered today to find out what each of them has(grade) in each class, whoever has the highest score on a subject gets a price from that class’s teacher. FIND OUT what each of them has in each class and who gets a price (for having the highest grades) from that subjects’ teacher.

M= Math S=Science H=History E=English P=P.E.

Max Jenny Ray Nancy 88.9% 87.5%H 94.5%P 90.7%M M 98.2%H 92.5%E 94.6%H 97.9%E 87.6%P 96.3%P 86.9%M 100%S 99.8%S 97.7%S 95.9%E 93.8%P 100%E 99.9%M 97.4%S 98.8%H Answer:

Math Scienc History Englis P.E. e h Max 88.9% 99.8% 98.2% 100% 87.6% Jenny 99.9% 97.7% 87.5% 92.5% 96.3% Ray 86.9% 97.4% 94.6% 95.9% 94.5% Nancy 90.7% 100% 98.8% 97.9% 93.8%

For Math Jenny has the highest score: 99.9%

For Science Nancy has the highest score: 100%

For History Nancy has the highest score: 98.9%

For English Max has the hieghest score: 100%

For P.E. Jenny has the hieghest score: 96.3%

4. Application (Explain how, if you were looking at a word problem, you would know that this skill could help you solve the problem)

I would know whether to use a matrix on a word problem or not if the problem had a lot of numbers along with names and subjects and was asking me for various specific things then I would much rather organize the data give then to spend a lot of time trying to find where every little piece of information that I need is.

5. Description of the process (Describe the process in a list or paragraph form. Be sure to explain the purpose of each step)

 Shows some of the basic vocabulary that is used when solving a matrix including the way in which it is done (with examples).

 This explains how a problem could be solved with a matrix and when the right time ofr using this skill would help.