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Algebra 2 Unit 3 – Chapter 4

Algebra 2 Unit 3 – Chapter 4. Section 4.1 – Matrix Operations Day 1. STEP 1. Rewrite the system as a linear system in two variables. 4 x + 2 y + 3 z = 1. Add 2 times Equation 3. 12 x – 2 y + 8 z = –2. to Equation 1. CONCEPT REVIEW. Solve the system.

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Algebra 2 Unit 3 – Chapter 4

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  1. Algebra 2Unit 3 – Chapter 4 Section 4.1 – Matrix Operations Day 1

  2. STEP 1 Rewrite the system as a linear system in two variables. 4x + 2y + 3z = 1 Add 2 times Equation 3 12x – 2y + 8z = –2 to Equation 1. CONCEPTREVIEW Solve the system. 4x + 2y + 3z = 1 Equation 1 2x – 3y + 5z = –14 Equation 2 Equation 3 6x – y + 4z = –1 SOLUTION 16x + 11z = –1 New Equation 1

  3. Add – 3 times Equation 3 to Equation 2. –18x + 3y –12z = 3 STEP 2 Solve the new linear system for both of its variables. Add new Equation 1 –16x – 7z = –11 and new Equation 2. CONCEPTREVIEW 2x – 3y + 5z = –14 –16x – 7z = –11 New Equation 2 16x+ 11z = –1 4z = –12 z = –3 Solve for z. x = 2 Substitute into new Equation 1 or 2 to find x.

  4. Substitute x = 2 and z = – 3 into an original equation and solve for y. STEP 3 CONCEPTREVIEW 6x–y + 4z = –1 Write original Equation 3. 6(2) –y + 4(–3) = –1 Substitute 2 for xand –3 for z. y = 1 Solve for y.

  5. WHAT IF THERE WAS A DIFFERENT WAY TO WRITE AND SOLVE THREE VARIABLE THREE EQUATION SYSTEMS? QUESTION OF THE DAY… ANSWER: MATRICES (plural of a matrix)

  6. What is a matrix?

  7. SECTION 4.1 – MATRICES Matrix – a rectangular arrangement of numbers into rows and columns. Dimensions – tell the number of rows and columns of a matrix, and it is how we define the size of a matrix.

  8. SECTION 4.1 – MATRICES Elements/Entries – the numbers that are located in a matrix. Equal Matrices – when two matrices have identical dimensions and identical corresponding elements/entries.

  9. Matrix Dimension & Size ROWS NAME COLUMNS 2 x 3 Matrix

  10. Parts of a Matrix Labeling Elements

  11. –1 4 2 0 3 0 –5 –1 3 + (–1) 0 + 4 –5 + 2 –1 + 0 2 4 –3 –1 = + = a. 7 4 0 -2 -1 6 -2 5 3 -10 -3 1 7 – (–2) 4 – 5 0 – 3 –2 – (–10) –1 – (–3) 6 – 1 9 –1 –3 8 2 5 b. = – = EXAMPLE 1: Add and subtract matrices Perform the indicated operation, if possible.

  12. 7 4 0 -2 -1 6 -2 5 3 -10 c. – EXAMPLE 1: Add and subtract matrices (cont.) Perform the indicated operation, if possible. NOT POSSIBLE; To add or subtract matrices the dimensions of the matrices must be equivalent. Here we have a 2 x 3 and a 2 x 2. Therefore

  13. –2(4) –2(–1) –2(1) –2(0) –2(2) –2(7) –8 2 –2 0 –4 –14 4 –1 1 0 2 7 a. –2 = = 4(–2) 4(–8) 4(5) 4(0) –2 –8 5 0 –3 8 6 –5 –3 8 6 –5 + + = b.4 –8 –32 20 0 –3 8 6 –5 + = EXAMPLE 2: Multiply a matrix by a scalar Perform the indicated operation, if possible.

  14. –8 + (–3) –32 + 8 20 + 6 0 + (–5) = –11 –24 26 –5 = EXAMPLE 2: Multiply a matrix by a scalar

  15. 1. –3 1 –5 –2 –8 4 –2 5 11 4 –6 8 + ANSWER GUIDED PRACTICE Perform the indicated operation, if possible. –5 6 6 2 –14 12

  16. 3. 2. 2 –1 –3 –7 6 1 –2 0 –5 –4 0 7 –2 –3 1 2 2 –3 0 5 –14 – – 4 ANSWER ANSWER GUIDED PRACTICE –6 –2 10 –2 –8 15 –8 4 12 28 –24 –4 8 0 20

  17. 4. 3 –3 –2 1 –2 –2 0 6 4 –1 –3 –5 3 + GUIDED PRACTICE ANSWER

  18. Solve a multi-step problem EXAMPLE 3 Manufacturing A company manufactures small and large steel DVD racks with wooden bases. Each size of rack is available in three types of wood: walnut, pine, and cherry. Sales of the racks for last month and this month are shown below.

  19. Last Month (A) This Month (B) Small Large Small Large Walnut Pine Cherry 95 114 316 215 205 300 125 100 278 251 225 270 Solve a multi-step problem EXAMPLE 3 Organize the data using two matrices, one for last month’s sales and one for this month’s sales. Then write and interpret a matrix giving the average monthly sales for the two month period. SOLUTION STEP 1 Organize the data using two 3 X 2 matrices, as shown.

  20. STEP 2 Write a matrix for the average monthly sales by first adding A and B to find the total sales and then multiplying the result by . 1 2 125 100 278 251 225 270 95 114 316 215 205 300 1 1 (A + B) = + 2 2 220 214 594 466 430 570 1 = 2 Solve a multi-step problem EXAMPLE 3

  21. 110 107 297 233 215 285 = Solve a multi-step problem EXAMPLE 3 STEP 3 Interpret the matrix from Step 2. The company sold an average of 110 small walnut racks, 107 large walnut racks, 297 small pine racks, 233 large pine racks, 215 small cherry racks, and 285 large cherry racks.

  22. Solve the matrix equation for x and y. –21 15 3 –24 5x –2 6 –4 3 7 –5 –y 3 + = –21 15 3 –24 5x –2 6 –4 3 7 –5 –y + = 3 Solve a matrix equation EXAMPLE 4 SOLUTION Simplify the left side of the equation. Write original equation.

  23. –21 15 3 –24 5x + 3 1 5 –4 – y 3 = 15x + 9 15 3 –12 – 3y –21 15 3 –24 = 15x + 9 = –21 –12 – 3y = 224 x = –2 y = 4 ANSWER The solution is x = –2 and y = 4. Solve a matrix equation EXAMPLE 4 Add matrices inside parentheses. Perform scalar multiplication. Equate corresponding elements and solve the two resulting equations.

  24. –30 14 38 –36 –20 30 GUIDED PRACTICE 5. In Example 3, find B – A and explain what information this matrix gives. ANSWER The difference in the number of DVD racks sold this month compare last month.

  25. GUIDED PRACTICE 12 10 2 –18 –3x –1 4 y 9 –4 –5 3 6. Solve –2 + = for x and y. ANSWER x = 5 and y = 6

  26. HOMEWORK Page 203-204 #11 – 35 ODD #37 – 41 ALL

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