Friday 10/11/13 Substitution **PRE AP - Unit 5: 3 by 3 Systems and MATRICES Name Per NOTES 1: Solve 3x3 Systems Ex. x 3y 2z 4 2x 5y 19 x 3 OYO 3x 2y + 4z 19 3y + z 3 z 3 1
Tuesday 10/15/13 Elimination NOTES 2: Solve 3x3 Systems Steps for Solving a 3 by 3 System 1. Write out the 3 equations (with 3 variables) 2. Eliminate 1 variable 3. Use the equation that you DID NOT JUST USE with 1 of the equations JUST USED and eliminate the SAME variable. 4. Use the 2 equations you developed as a 2 by 2 system to solve for the remaining variables 5. Substitute that value back into an equation from the 2 by 2 system and solve for the other variable 6. Substitute both variables into an original equation (3 by 3) to solve for the last variable 7. Check your answer by plugging into each equation from the original system and checking that it makes a true statement! Ex. x + y z 0 x + 2y + 2z 22 6x 3y + z 6 OYO 2x y + 2z 4 x + y 2z 5 3x y 2z 15 2
Block Day 10/16-17/13 NOTES 3 Part I: MATRICES Vocab, Properties, Determinants & Inverse Matrix Dimensions DETERMINANTS denoted by - det or The matrix must be BY HAND det a c b d Example 1: 3 4 5 3 IN THE CALCULATOR 3 2 [B],det[B] 4 6 Example 3: [D] 4 2 0 3 10 1 2 6 1, D Example 4: [c] 4 2 3,det[c] 0 1 4 [A] 2 3 4 5 1 2, A 10 3 1 3
Block Day 10/16-17/13 NOTES 3 Part II: INVERSES OF MATRICES If A a b c d, then A-1 1 d b det A c a 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 [B] 3 2,det[B] 4 6 Example 1: [B] 1 7 14 [ D],det[ D] 3 6 [D] 1 [c] 6 11,det[c] 4 7 [c] 1 [A] 1 0 2 1 1 6 3 [A] -1 4
Block Day 10/16-17/13 NOTES 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 0 4 1 2 3 3 + 0 2 1 1 2 0 2 3 5 1 0 7 + 6 9 2 3 8 2 6 3 2 5 1 6 Example 3: 5 2 3 2 7 3 1 7 8 4 0 2 6 2 1 8 11 5 3 9 3 7 8 7 1 3 8 8 2 4 7 9 10 5
Block Day 10/16-17/13 NOTES 3 Part IV: MATRICES SCALAR MULTIPLICATION Multiply by a Scalar each entry is multiplied by the number called a Example 1: Find 3G when 0 4 1 3G 3 0 2 1 G 0 4 1 0 2 1 5 0 2 1 1 1 2 3 4 0 2 4 4 3 5 COMBINATION PROBLEMS: Use the order of operation (PEMDAS) [Q] 2 3 0 1 [P] 4 1 6 2 Example 1: 2[Q]-[P] 2([P]-[Q]), 6
Block Day 10/16-17/13 NOTES 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: x 7 12 7 15 3 15 3 4x 1 16 1 3 y 3 3 10 Example 3: Example 5: Example 6: 2x 7 0 15 0 5 y + 4 5 12 9x 7 3y 7 15 14 + 3 4x 14 y 17 8 3x + 3 3 4y 11 30 14 + 5 3x 2 12y + 4 1 3 2x 8 18 8 3 y 3 5 x + 4 3 16 3 9 y 3 9 7 x 3 2 4 2 y + 5 3 3 3 x 2 3y 4 16 2 10 3x + 7 y 17 0 3x 6 13 y 4 7 26 20 + 6 5x + 1 7 4 5y 13 25 7
Friday 10/18/13 NOTES 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 2 3 3 8 4 3 5 2 3 3 8 4 3 5 7 A B 9 5 2 0 6 4 1 2 C 4 1 D 9 5 2 0 6 0 1 6 7 2 5 8 0 1 6 7 2 [A] [B] YES it can multiply [C] [D] No it can t multiply 2 3 3 5 3 2 3 5 Calculator NOTE: If the 1 st matrix does not have the same number of columns as rows of the 2 nd 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 0 1 2 1 3 1 4 0 3 5 2 3 Wouldn t it be nice to have a CALCULATOR do this for us? WRITE OUT THE DIMENSIONS FIRST TO CHECK THAT IT WILL WORK!!!!! 2 3 2 1 0 5 2 1 4 3 0 1 2 3 1 8 5 2 0 7 3 4 2 2 3 0 1 3 8
Monday 10/21/13 NOTES 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] The inverse matrix is used to cancel out other matrices. Whatever is done to one side of the equation must be done to the other so to get rid of [A] we multiply by [A] -1 on both sides. [A][x] [B] would become (this can be memorized) 4 1 x 3 6 2 y 5 would become 1 x 4 1 3 y 6 2 5 Example 1: Solve for x and y given the following matrices 2x 1 8 1 6 7 + y 6 21 x 6y + 4z 12 x + y 4z 12 2x + 2y 5z 15 This system must be solved for the constant before it can be written in matrix form! Example 3: y x + z + 5 z 3y 3 2x y 4 3x 2y + 4z 19 3y + z 3 z 3 2x y + 2z 4 x + y 2z 5 3x y 2z 15 9
Tuesday 10/22/13 NOTES 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? 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 4 children s books and 8 painting books. The second day she earned $14 selling 4 painting books and 4 boxes of markers. The third day she earned $16 selling 10 children s books. If she sells 5 children s books and 1 box of markers the fourth day, how much will she make? 10
Tuesday/Block 10/22/13 MATRICES REVIEW TEST IS ON BLOCK DAY 2 nd Half of Class!!!!! STILL NEED!! 11