PRE AP - Unit 5: 3 by 3 Systems and MATRICES

**PRE AP - Unit 5: 3 by 3 Systems and MATRICES

Name______Per ______

Friday 10/11/13NOTES 1: Solve 3x3 Systems

Substitution

Ex.

OYO

Ex.

Tuesday 10/15/13NOTES 2: Solve 3x3 Systems

Elimination

Steps for Solving a 3 by 3 System

Write out the 3 equations (with 3 variables)
Eliminate 1 variable
Use the equation that you DID NOT JUST USE with 1 of the equations JUST USED and eliminate the SAME variable.
Use the 2 equations you developed as a 2 by 2 system to solve for the remaining variables
Substitute that value back into an equation from the 2 by 2 system and solve for the other variable
Substitute both variables into an original equation (3 by 3) to solve for the last variable
Check your answer by plugging into each equation from the original system and checking that it makes a true statement!

Ex.

OYO

Block Day 10/16-17/13NOTES 3 Part I: MATRICES – Vocab, Properties, Determinants Inverse

Matrix –

Dimensions –

DETERMINANTS

denoted by - det or

The matrix must be

BY HAND

Example 1:

IN THE CALCULATOR

Example 2:

Example 3: [D] = , |D| =

Example 4:

OYO: [A] = , |A| =

Block Day 10/16-17/13NOTES 3 Part II:INVERSES OF MATRICES

If A = , then A-1 = when det A 0.

If the determinant is ______there will be ______inverse!!

Calculator NOTE: If you take the inverse when the determinant is zero because it would be dividing by zero  the calculator will give you ERR:SINGULAR MAT – meaning you have a singular matrix

Example 1:

Example 2:

OYO:

OYO:[A] =

[A]-1 =

Block Day 10/16-17/13NOTES 3 Part III:MATRICES –Add Subtract

Add/ Subtract Matrices – Both matrices must have

Calculator NOTE: If the matrices do not have the same dimensions because it would be each entry must have a partner to add/subtract to the calculator will give you ERR:DIM MISMATCH – meaning your dimensions don’t match and you cannot perform this operation (+/-/×/÷)

Example 1: G + H

Example 2:

Example 3:

OYO:

Block Day 10/16-17/13NOTES 3 Part IV: MATRICES – SCALAR MULTIPLICATION

Multiply by a Scalar – each entry is multiplied by the number called a ______

Example 1:Find 3G when

Example 2:

OYO:

COMBINATION PROBLEMS: Use the order of operation (PEMDAS)

[Q] = [P] =

Example 1:2[Q]-[P]

OYO: 2([P]-[Q]),

Block Day 10/16-17/13NOTES 3 Part V: MATRICES – Solving for Variable in Matrices

When solving for a variable in a matrix, ______entries are equal.

Set up an equation(s) for each example and solve.

Example 1:

Example 2:

Example3:

Example 5:

Example 6:

OYO:

Friday 10/18/13NOTES 4: MATRIX MULTIPLICATION

Remember: we write the dimensions of matrices as r × c

# columns of [A] = # rows of [B] so that you can match up the entries to complete the matrix multiplication

[A] [B]YES – it can multiply[C] [D]No – it can’t multiply

2 × 3=3 ×53×2≠3 ×5

Calculator NOTE:If the 1st matrix does not have the same number of columns as rows of the 2nd matrix there would not be a partner multiply with  the calculator will give you ERR:DIM MISMATCH – meaning your dimensions don’t match and you cannot perform this operation (+/-/×/÷)

Example 1: AB

Wouldn’t it be nice to have a CALCULATOR do this for us? 

WRITE OUT THE DIMENSIONS FIRST TO CHECK THAT IT WILL WORK!!!!!

Example 2:

OYO:

Monday 10/21/13NOTES 5: MATRICES – SOLVE MATRIX PROBLEMS

We use the ______matrix to solve equations.

[A] [x] = [B]

[A]-1∙ [A] [x] = [A]-1∙ [B]

[x] = [A]-1∙ [B]

[A][x] = [B] would become ______(this can be memorized)

would become

Example 1: Solve for x and y given the following matrices

Example 2:

☼ This system must be solved for the constant before it can be written in matrix form!

Example 3:

OYO:

Tuesday 10/22/13NOTES 6: MATRICES – SOLVE MATRIX WORD PROBLEMS

Example 1: A movie charges $5 for an adult ticket and $2 for a child ticket. The theater sold 785 tickets for $3280. How many adult tickets and how many child tickets were sold?

Example 2: A company makes 3 types of cables. Type A requires 3 black, 3 white, and 2 red wires. Type B requires 1 black, 2 white, and 1 red wires. Type C requires 2 black, 1 white, and 2 red. They used 100 black, 110 white and 80 red wires. How many of each cable were made?

Example 3: Titan inherited $50,000 and invested part of it in a money market account, part in municipal bonds, and part in a mutual fund. After one year, he received a total of $3,580 in simple interest from the three investments. The money market paid 6% annually, the bonds paid 7% annually, and the mutually fund paid 8% annually. There was $10,000 more invested in the bonds than the mutual funds. Find the amount John invested in each category.

Example 4: Ella had a supply sale to raise money for a charity. The first day she earned $20 selling 4children’s books and 8painting books. The second day she earned $14 selling 4painting books and 4boxes of markers. The third day she earned $16 selling 10children’s books. If she sells 5children’s books and 1 box of markers the fourth day, how much will she make?

Tuesday/Block 10/22/13 MATRICES – REVIEW – TEST IS ON BLOCK DAY – 2nd Half of Class!!!!!

STILL NEED!!