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Section 4.1 – Matrix Operations Day 1

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

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

3 Solve the new linear system for both of its variables.
CONCEPT REVIEW 2x – 3y + 5z = –14 Add – 3 times Equation 3 to Equation 2. –18x + 3y –12z = 3 –16x – 7z = –11 New Equation 2 STEP 2 Solve the new linear system for both of its variables. 16x + 11z = –1 Add new Equation 1 and new Equation 2. –16x – 7z = –11 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
CONCEPT REVIEW STEP 3 Substitute x = 2 and z = – 3 into an original equation and solve for y. 6x – y + 4z = –1 Write original Equation 3. 6(2) – y + 4(–3) = –1 Substitute 2 for x and –3 for z. y = 1 Solve for y.

5 ANSWER: MATRICES (plural of a matrix)
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)

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7 What is a matrix?

8

9 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.

10 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.

11 Matrix Dimension & Size
ROWS NAME COLUMNS 2 x 3 Matrix

12 Parts of a Matrix Labeling Elements

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

14 EXAMPLE 1: Add and subtract matrices (cont.)
Perform the indicated operation, if possible. 0 -2 c. 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

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

16 EXAMPLE 2: Multiply a matrix by a scalar
–8 + (–3) –32 + 8 (–5) = –11 –24 –5 =

17 GUIDED PRACTICE Perform the indicated operation, if possible. ANSWER + 4 – 1. – –5 –2 –8 4 2 –

18 GUIDED PRACTICE 7 –2 5 –14 2. ANSWER – –2 –2 ANSWER 3. 2 –1 –3 – –5 – 4 28 – –4

19 GUIDED PRACTICE 4 –1 –3 –5 –2 –2 4. 3 + ANSWER 3 –3

20 EXAMPLE 3 Solve a multi-step problem 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.

21 EXAMPLE 3 Solve a multi-step problem 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. Walnut Pine Cherry Last Month (A) This Month (B) Small Large Small Large

22 EXAMPLE 3 Solve a multi-step problem 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 (A + B) = 1 2 + 1 2 =

23 EXAMPLE 3 Solve a multi-step problem = 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.

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

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

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

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

28 HOMEWORK Page #11 – 35 ODD #37 – 41 ALL


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