**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:
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:
OYO:
OYO:
OYO:
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:
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:
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!!
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