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Warm-up 1.2

- Find matrix G if G =
[0 2] [0 -3] [-1 0]

[1 3] + [-1 4] - [2 -2]

[2 1] [2 2] [-3 1]

Reminders:Sorry!

I’ll be absent tomorrow for a training I have to go to.

I will leave specific sub plans.

All work must be turned in (I will come by the school and pick up work to grade it).

If you have any questions, comment on the website or leave a note on your paper

*What level are you?

§1.2: Organizing Data & Matrices

LEQ: How can statistical data be organized into matrices?

Uses for matrices…

Topics

- Organizing Statistical Data
- What is the difference between a 2x3 matrix and a 3x2 matrix?
- Presenting Data from a table in a matrix
- Elements, rows, columns (row, column)
- Equal matrices = iff same dimensions and corresponding elements are equal
- Solving matrices with algebra involved

Adding and Subtracting Matrices

LEQ: How do you add, subtract, and multiply matrices?

- Matrix addition/subtraction – add/subtract corresponding elements
- Explain why you add matrices only if they have the same dimensions.
- Is matrix subtraction commutative? Why?
- Matrix equation – addition/subtraction properties of equality

10.3: Matrix Multiplication

How are the dimensions of a matrix related to the ability to multiply matrices?

Scalar Multiplication (Scalar)

You can also multiply two matrices.

Method:

- Multiply the elements of each row of the first matrix by the elements of the first column of the second matrix.
- Add the products.

Warm-up 1/11/08

Find the product, if possible. If not, possible, write produce undefined.

- [2] [3 1 -2]
[1]

- [1] [1 0]
[2] [0 1]

- [-1 0 -2] [1]
[2 3 1] [3]

[2]

Identity and Inverse Matrices

LEQ: How would you describe the identity matrix and its uses?

A square matrix is a matrix with the same number of columns as it has rows.

The Identity matrix is a square matrix with 1’s along the diagonal and 0’s everywhere else.

Inverse Matrix

- IF X is the inverse matrix of A, then
AX = I (A times its inverse = the identity)

Not all matrices have inverses.

If detA = 0, then A does not have an inverse.

If detA ≠ 0, then A does have an inverse.

[ c d]

Then, A-1 = *swap a & c, make b,d neg.

1 [c -b]

(ad – bc) [a -d]

You can also use inverse matrices to solve matrix equations.

[0 -4] X = [0]

[0 -1] [4]

For this problem A-1 does not exist, so you cannot solve the problem.

Examples

- 3-6 Worksheet

Solving Systems of LinearEquations in Three VariablesUsing the Elimination Method

- Note that there is more than one way that you can solve this type of system. Elimination (or addition) method is one of the more common ways of solving the problem by algebra, so I choose to show it this way.

- Step 1: Simplify and put all three equations in the form Ax + By + Cz = D if needed .
- This would involve things like removing ( ) and removing fractions.
- To remove ( ): just use the distributive property.
- To remove fractions: since fractions are another way to write division, and the inverse of divide is to multiply, you remove fractions by multiplying both sides by the LCD of all of your fractions.

- Step 2: Choose to eliminate any one of the variables from any pair of equations.
- This works in the same manner as eliminating a variable with two linear equations and two variables
- At this point, you are only working with two of your equations. In the next step you will incorporate the third equation into the mix.

- Looking ahead, any pair of equations.you will be adding these two equations together.
- Make sure one of the variables cancels out.

- It doesn't matter which variable you choose to drop out.
- For example, if you had a 2x in one equation and a 3x in another equation, you could multiply the first equation by 3 and get 6x and the second equation by -2 to get a -6x. So when you go to add these two together they will cancel out.

- Step 3: Eliminate the SAME variable chosen in step 2 from any other pair of equations, creating a system of two equations and 2 unknowns.
- Basically, you are going to do another elimination step, eliminating the same variable we did in step 2, just with a different pair of equations.
- Follow the same basic logic as shown in step 2

- Step 4: Solve the remaining system found in step 2 and 3, just as if it is a system of 2 equations in 2 variables
- After steps 2 and 3, there will be two equations and two unknowns which is just a system of 2 equations and 2 variables.
- You can use any method you want to solve it.
- When you solve this system that has two equations and two variables, you will have the values for two of your variables.

- Remember that just as if it is a system of 2 equations in 2 variables if both variables drop out and you have a FALSE statement, that means your answer is no solution.
- If both variables drop out and you have a TRUE statement, that means your answer is infinite solutions, which would be the equation of the line.

- Step 5: Solve for the third variable. just as if it is a system of 2 equations in 2 variables
- If you come up with a value for the two variables in step 4, that means the three equations have one solution. Plug the values found in step 4 into any of the equations in the problem that have the missing variable in it and solve for the third variable.

- Step 6: Check. just as if it is a system of 2 equations in 2 variables
- You can plug in the proposed solution into ALL THREE equations. If it makes ALL THREE equations true then you have your solution to the system.
- If it makes at least one of them false, you need to go back and redo the problem.

- There are three possible outcomes that you may encounter when working with these systems:
- one solution
- no solution
- infinite solutions

Solve for x, y, and z when working with these systems:

-3x + 8y + 1z = -18

-2x – 6y – 9z = -15

-6x + 5y + 3z = 13

(-3,-4,5)

Solving systems using Matrices when working with these systems:

Ex. [A]x = [C]

Typically, you would divide both sides by [A]

Division with matrices is notated as [A]-1.

So, [A]-1[A] = I

Do the same thing to the other side [A]-1[C]

Remember multiplication is not commutative, thus, you always do the order the same way!

Assignment when working with these systems:

#1-3 on back of WS

Complete Worksheet

(you may use calculators)

HW: Take home quiz

Warm-up 1/14/08 when working with these systems:

Grapheach of theseinequalities:

- 2x – 3y<12
- x + 5y < 20

Solving Systems of Linear Inequalities: when working with these systems:

LEQ: How do you graph systems of inequalities?

Graph equations

Shade region

Ex) when working with these systems:

- y> ( 2/3)x – 4 y< (– 1/5)x + 4 x > 0
- The solution is where all the inequalities work (the region where all three individual solution regions overlap)
- In this case, the solution is the shaded part in the middle

- Some systems may have regions that are completely bounded when working with these systems:
- Other systems will have regions that are unbounded in one direction
- Still other systems will have no common solution region

LEQ: How do you find the maximum or minimum value of a function defined for a polygonal convex set?

Linear programming is the process of taking various linear inequalities relating to some situation, and finding the "best" value obtainable under those conditions.

- In "real life", linear programming is part of a very important area of mathematics called "optimization techniques".
- This field of study (or at least the applied results of it) are used every day in the organization and allocation of resources.
- These "real life" systems can have dozens or hundreds of variables, or more.
- In algebra, though, you'll only work with the simple (and graphable) two-variable linear case.

- The general process is to graph the inequalities (which, in this context, are called "constraints")
- Form a walled-off area on the x,y-plane (which is called the "feasibility region").
- Figure out the coordinates of the corners of this feasibility region (that is, the intersection points of the various lines),
- Test these corner points in the formula (called the "optimization equation") for which you're trying to find the highest or lowest value.

Find the maximal and minimal value of this context, are called "constraints") z = 3x + 4y subject to the following constraints:

- Find the maximal and minimal value of z = 3x + 4y subject to the following constraints:

To solve this context, are called "constraints")

- Graph the system
- Find the intersection points
- The maximum and minimum will be at the corner points
- Plug in the important points you found to find the minimum and maximum values

Assignment this context, are called "constraints")

Graphing Systems of Inequalities

Finding minimum and maximum

values of a function.

Warm-up 1/15/08 this context, are called "constraints")

Solve each system of equations or inequalities.

- x + 2y = 8
-x – 3y = 2

- 9x + 12y < 36
34x – 17y > 34

- Given the inequalities, linear-programming exercises are pretty straightforward, if sometimes a bit long.
- The hard part is usually the word problems, where you have to figure out what the inequalities are. So I'll show how to set up some typical linear-programming word problems.

- At a certain refinery, the refining process requires the production of at least two gallons of gasoline for each gallon of fuel oil. To meet the anticipated demands of winter, at least three million gallons a day will need to be produced. The demand for gasoline, on the other hand, is not more than 6.4 million gallons a day. If gasoline is selling for $1.90 per gallon and fuel oil sells for $1.50/gal, how much of each should be produced in order to maximize revenue?

- The question asks for the number of gallons which should be produced, so I should let my variables stand for "gallons produced".
- x: gallons of gasoline producedy: gallons of fuel oil produced

- Since this is a "real world" problem, I know that I can't have negative production levels, so the variables can't be negative. This gives me my first two constraints: namely, x> 0 and y> 0.

- Since I have to have at least two gallons of gas for every gallon of oil, then x> 2y. For graphing, of course, I'll use the more manageable form "y< ( 1/2 )x".

- The winter demand says that gallon of oil, then y> 3,000,000; note that this constraint the eliminates the need for the "y> 0" constraint. The gas demand says that x< 6,400,000.

- I need to maximize revenue gallon of oil, then R, so the optimization equation is R = 1.9x + 1.5y. Then the model for this word problem is as follows:
- R = 1.9x + 1.5y , subject to:x> 0x< 6,400,000y> 3,000,000y< ( 1/2 )x

- When you test the corner points at (6.4m, 3.2m), (6.4m, 3m), and (6m, 3m), you should get a maximal solution of R = $16.96m at (x, y) = (6.4m, 3.2m).

Test Expectations: and (6m, 3m), you should get a maximal solution of

- Fewer than 20 problems
- 30% word problems
- 60% regular problems
- 10% optimization/application problems

Warm-up 1/16/08 and (6m, 3m), you should get a maximal solution of

Create a system of 3 equations to solve the problem:

James sold magazine subscriptions with three prices: $20, $11, and $28. He sold 3 fewer of the $20 subscriptions than of the $11 subscriptions and sold a total of 37 subscriptions. If his total sales amounted to $735, how many $28 subscriptions did James sell?

Solve: and (6m, 3m), you should get a maximal solution of

Tasty Bakery sells three kinds of muffins: chocolate chip muffins at 25 cents each, oatmeal muffins at 30 cents each, and blueberry muffins at 35 cents each. Charles buys some of each kind and chooses twice as many blueberry muffins as chocolate chip muffins. If he spends $4.35 on 14 muffins, how many of each did he buy?

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