Warm-Up: May 7, 2015

Investigation A Questions?

Matrix AlgebraHonors Appendix Investigation B

Advanced Integrated Math I

You-Try #2

You-Try #3

You-Try #4

You-Try #5

You-Try #6

Assignment

• Page 721 #8-11, 14

Page 721

Warm-Up: May 8, 2015

Homework Questions?

May 8, 2015 – Quiz Time

• Clear everything off of your desk except pencils, erasers, a ruler, and a non-graphing calculator.• If you appear to be talking, looking at anyone else’s quiz, allowing

another student to look at your quiz, or using any unauthorized aide, you will receive a zero.• 30 minute time limit (except for students with an IEP or 504)

Basic Matrix Operations – Addition, Subtraction,

and Scalar MultiplicationSection H.07

Advanced Integrated Math I

Matrices

• A rectangular (two-dimensional) array of numbers/variables/expressions is called a matrix.• An array is an ordered arrangement.

• The plural of matrix is matrices.• A matrix is like a table without the lines or labels.

Matrix Addition and Subtraction

• Only matrices of the same size can be added or subtracted.• Add/subtract the corresponding entries.

Warm-Up: May 11, 2015 • Use the matrices above to compute each of the following:a. b. c. d.

Scalar Multiplication

• Scalar multiplication is when a matrix is multiplied by a constant.• The constant is a scalar.

• Multiply every element of the matrix by the scalar.

Matrix Size• Matrices have rows and columns.• Rows go side to side.• Columns go up and down.

• A matrix with rows and columns is an matrix.

You-Try #1

You-Try #2

• Let with .a. Write out if it is .b. Write out if it is .

Assignment

• Read Section H.07 (pages 724-727)• Page 729 #8-14, 16-20

Page 729

Warm-Up: May 12, 2015• Find the dot product of and . (Hint: the answer is a scalar.)

Homework Questions?

Dot ProductSection H.08

Advanced Integrated Math I

Dot Product

• The dot product of two or matrices is

• or matrices are called vectors or n-tuples.

You-Trys

You-Trys

Warm-Up: May 13, 2015• Compute each of the following dot products, or explain why they are

undefined.1. 2. 3. 4.

Minds in Action• Tony and Sasha have been asked to justify, or find counterexamples

to, two proposed rules for n-tuples. and

• Tony: Hey Sasha, what’s there to show? This is just basic algebra. Everybody knows that and . These are just the basic rules of associativity and commutativity.• Sasha: Yes, we already know them for real-number addition and

multiplications, but here we have -tuples and dot products.• Tony: Ugh! If all the rules are the same and we know them, I don’t

see why we should have to show them again.

Minds in Action• Sasha: Well, they aren’t really the same, because -tuples aren’t

numbers, so their sum isn’t quite a number sum. Maybe the rules aren’t even true. Though I hope they are, because their form is so familiar – something we would want to use. So how should we start?• Tony: I still don’t get it. What do you mean that the sum isn’t a

number sum?• Sasha: Well, when we say for numbers, we are saying and old stuff

like that. But when we say for -tuples, we are saying , and new stuff like that. This is a new use of the plus sign, so we’ve got to think about whether it’s right. So the first issue is, what does the plus sign mean for -tuples? Our teacher never really told us.

and

Minds in Action• Tony: If they’re just thin matrices, then it should be the same

meaning as for matrices: add corresponding entries.• Sasha: So and would have to be the same size, say , and we could

write

Those old indices again, but only one at a time, thank goodness.• Tony: is . Each entry is .• Sasha: Hey, each entry of is . I think we’re done, because .

and

Minds in Action• Tony: No way! You’re going in circles. You’re saying and commute

because and commute.• Sasha: No, I’m not going in circles, because and are -tuples and and

are numbers. We already know that numbers commute. We’ve reduced it to the previous case!• Tony: Okay, I see what you mean. I guess they are different. It still

feels like we’re just doing the same thing, but let’s go on to the other rule.• Sasha: Well, I see an important difference here. Both sides of are

numbers. Both sides of are thin matrices.

and

Minds in Action• Tony: Good news. To show that two matrices are equal, we had to

show that they were equal entry by entry. Now we just have two numbers that have to be equal.• Sasha: Yeah, but each number is constructed in a complicated way.

On the left, , we start with . Suppose . Then we multiply by some number . Then we dot with some other 3-tuple . Let’s say .• Tony: Who says these vectors are 3-tuples?• Sasha: No one, but if we see what is going on for 3-tuples, maybe the

general pattern won’t be much different. Try it. You compute .

and

Minds in Action• Tony: All right. , so then would be +• Sasha: Thanks. Now the right side of the rule, , is . So are the

two sides the same or not?

and

Think-Pair-Share

• Are the two expressions that Tony and Sasha found actually the same?

and

Assignment

• Read Section H.08 (pages731-734)• Page 736 #11-14, 16-20

Page 736

Warm-Up: May 14, 2015

• With your partner, come up with an algorithm to multiply matrices.

Homework Questions?

Matrix MultiplicationSection H.09

Advanced Integrated Math I

Matrix Multiplication

• In order to multiply , must have as many columns as has rows.• If is and is , then is .• Let the rows of be called .• Let the columns of be called .

Example 1

Matrix Multiplication

You-Trys

Matrices and Systems of Linear Equations

You-Try (p.746 #5)

You-Trys

Assignments

• Read Section H.09 (pages 739-744)• Page 748 #15, 17-19, 21, 25-27

• Page 753 #1-8

Page 748

Warm-Up: May 16, 2015

• Compute

Homework Questions?

Matrices on TI-84

Warm-Up, revisited

• Compute

Example

• Solve the following system of equations: