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Organizing Data Into Matrices

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  1. Organizing Data Into Matrices Lesson 4-1 Check Skills You’ll Need (For help, go to Skills Handbook page 842.) Use the table at the right. • How many units were imported to the • United States in 1996? • How many were imported in 2000? U.S. Passenger Vehicles and Light Trucks Imports And Exports (millions) 4.678 million units 1996 1998 2000 Imports 4.678 5.185 6.964 Exports 1.295 1.331 1.402 Source: U.S. Department of Commerce. 6.964 million units Check Skills You’ll Need 4-1

  2. Organizing Data Into Matrices Lesson 4-1 Additional Examples Write the dimensions of each matrix. 7 –4 12 9 The matrix has 2 rows and 2 columns and is therefore a 2  2 matrix. a. The matrix has 1 row and 3 columns and is therefore a 1  3 matrix. b. 0 6 15 Quick Check 4-1

  3. 3 –1 –8 5 1 8 4 9 8 –4 7 –5 3 –1 –8 5 1 8 4 9 8 –4 7 –5 a. K = b. K = k12 is the element in the first row and second column. k32 is the element in the third row and second column. Organizing Data Into Matrices Lesson 4-1 Additional Examples Identify each matrix element. 3 –1 –8 5 1 8 4 9 8 –4 7 –5 K = a.k12 b.k32 c.k23 d.k34 Element k12 is –1. Element k32 is –4. 4-1

  4. 3 –1 –8 5 1 8 4 9 8 –4 7 –5 3 –1 –8 5 1 8 4 9 8 –4 7 –5 c. K = d. K = k23 is the element in the second row and third column. k34 is the element in the third row and the fourth column. Organizing Data Into Matrices Lesson 4-1 Additional Examples (continued) 3 –1 –8 5 1 8 4 9 8 –4 7 –5 K = a.k12 b.k32 c.k23 d.k34 Quick Check Element k23 is 4. Element k34 is –5. 4-1

  5. Ed X X Jo X Lew X X X X Ed Jo Lew Wins Losses 5 6 3 2 1 4 Organizing Data Into Matrices Lesson 4-1 Additional Examples Three students kept track of the games they won and lost in a chess competition. They showed their results in a chart. Write a 2  3 matrix to show the data. = Win X = Loss Let each row represent the number of wins and losses and each column represent a student. Quick Check 4-1

  6. Each column represents a different year. 1996 1998 2000 N = Import Exports 4.678 5.185 6.964 1.295 1.331 1.402 Each row represents imports and exports. Organizing Data Into Matrices Lesson 4-1 Additional Examples Refer to the table. U.S. Passenger Car Imports And Exports (millions) 199619982000 Imports 4.678 5.185 6.964 Exports 1.295 1.331 1.402 a. Write a matrix N to represent the information. Use 2  3 matrix. Source: U.S. Department of Commerce. 4-1

  7. Organizing Data Into Matrices Lesson 4-1 Additional Examples (continued) U.S. Passenger Car Imports And Exports (millions) 199619982000 Imports 4.678 5.185 6.964 Exports 1.295 1.331 1.402 b. Which element represents exports for 2000? Source: U.S. Department of Commerce. Exports are in the second row. The year 2000 is in the third column. Element n23 represents the number of exports for 2000. Quick Check 4-1

  8. Deposits Withdrawals A $450 $370 B $475 $289 C $364 $118 D $420 $400 A B C D Deposits Withdrawals 450 475 364 420 370 289 118 400 Organizing Data Into Matrices Lesson 4-1 Lesson Quiz 8 4 0 1 9 3 –5 0 –1 2 6 1 3  4 1. Write the dimensions of the matrix. M = 2. Identify the elements m24, m32, and m13 of the matrix M in question 1. 0, 2, 0 3. The table shows the amounts of the deposits and withdrawals for the checking accounts of four bank customers. Show the data in a 2  4 matrix. Label the rows and columns. 4-1

  9. Adding and Subtracting Matrices Lesson 4-2 Check Skills You’ll Need (For help, go to Skills Handbook page 845.) Simplify the elements of each matrix. 1. 2. 3. 4. 5. 6. 10 + 4 0 + 4 –2 + 4 –5 + 4 5 – 2 3 – 2 –1 – 2 0 – 2 –2 + 3 0 – 3 1 – 3 –5 + 3 3 + 1 4 + 9 –2 + 0 5 + 7 8 – 4 –5 – 1 9 – 1 6 – 9 2 + 4 6 – 8 4 – 3 5 + 2 Check Skills You’ll Need 4-2

  10. Adding and Subtracting Matrices Lesson 4-2 Check Skills You’ll Need Solutions 1. = 2. = 3. = 4. = 5. = 6. = 10 + 4 0 + 4 –2 + 4 –5 + 4 14 4 2 –1 5 – 2 3 – 2 –1 – 2 0 – 2 3 1 –3 –2 –2 + 3 0 – 3 1 – 3 –5 + 3 1 –3 –2 –2 3 + 1 4 + 9 –2 + 0 5 + 7 4 13 –2 12 8 – 4 –5 – 1 9 – 1 6 – 9 4 –6 8 –3 2 + 4 6 – 8 4 – 3 5 + 2 6 –2 1 7 4-2

  11. Evening C A Theater 1 54 439 Theater 2 58 386 Matinee Evening Theater C A C A 1 198 350 54 439 2 201 375 58 386 Matinee C A Theater 1 198 350 Theater 2 201 375 Adding and Subtracting Matrices Lesson 4-2 Additional Examples The table shows information on ticket sales for a new movie that is showing at two theaters. Sales are for children (C) and adults (A). a. Write two 2  2 matrices to represent matinee and evening sales. 4-2

  12. 198 350 201 375 54 439 58 386 198 + 54350 + 439 201 + 58375 + 386 + = C A Theater 1 252 789 Theater 2 259 761 = Adding and Subtracting Matrices Lesson 4-2 Additional Examples (continued) b. Find the combined sales for the two showings. Quick Check 4-2

  13. 9 + 0 0 + 0 –4 + 0 6 + 0 3 + (–3) –8 + 8 –5 + 5 1 + (–1) = = 9 0 –4 6 0 0 0 0 = = Adding and Subtracting Matrices Lesson 4-2 Additional Examples Find each sum. 9 0 –4 6 0 0 0 0 3 –8 –5 1 –3 8 5 –1 a. + b. + Quick Check 4-2

  14. Write the additive inverses of the elements of the second matrix. 4 8 –2 0 –7 9 –4 –5 A– B = A + (–B) = + 4 + (–7) 8 + 9 –2 + (–4) 0 + (–5) Add corresponding elements = –3 17 –6 –5 Simplify. = Adding and Subtracting Matrices Lesson 4-2 Additional Examples 4 8 –2 0 7 –9 4 5 A = and B = . Find A – B. Method 1: Use additive inverses. 4-2

  15. 4 8 –2 0 7 –9 4 5 A– B = – 4 – 7 8 – (–9) –2 – 4 0 – 5 Subtract corresponding elements = –3 17 –6 –5 Simplify. = Adding and Subtracting Matrices Lesson 4-2 Additional Examples (continued) Method 2: Use subtraction. Quick Check 4-2

  16. 2 5 3 –1 8 0 10 –3 –4 9 6 –9 X – = 2 5 3 –1 8 0 2 5 3 –1 8 0 10 –3 –4 9 6 –9 2 5 3 –1 8 0 2 5 3 –1 8 0 X – + = + Add to each side of the equation. 12 2 –1 8 14 –9 Simplify. X = Adding and Subtracting Matrices Lesson 4-2 Additional Examples Quick Check 2 5 3 –1 8 0 10 –3 –4 9 6 –9 Solve X – = for the matrix X. 4-2

  17. 17 5 4 – 10 –2 + 1 0 – 8 + 9 5 –6 –1 0 0.7 a.M = ; N = 79 17 5 4 – 10 –2 + 1 0 – 8 + 9 5 –6 –1 0 0.7 M = ; N = 7 9 7 9 Both M and N have three rows and two columns, but – 0.7. M and N are not equal matrices. = / Adding and Subtracting Matrices Lesson 4-2 Additional Examples Determine whether the matrices in each pair are equal. 4-2

  18. 8 0.2 27 9 16 4 – – 3–4 40–3 P = ; Q = 8 0.2 12 4 Adding and Subtracting Matrices Lesson 4-2 Additional Examples (continued) 27 9 16 4 – – 3 –4 40 –3 b.P = ; Q = 12 4 Both P and Q have two rows and two columns, and their corresponding elements are equal. P and Q are equal matrices. Quick Check 4-2

  19. 2m – n –3 8 –4m + 2n 15 m + n 8 –30 = for m and n. 2m – n–3 8 –4m + 2n 15m + n 8 –30 = 2m – n = 15–3 = m + n–4m + 2n = –30 Adding and Subtracting Matrices Lesson 4-2 Additional Examples Solve the equation Since the two matrices are equal, their corresponding elements are equal. 4-2

  20. 3m = 12 Add the equations. m = 4 Solve for m. Adding and Subtracting Matrices Lesson 4-2 Additional Examples (continued) Solve for m and n. 2m – n = 15 m + n = –3 4 + n = –3 Substitute 4 for m. n = –7 Solve for n. The solutions are m = 4 and n = –7. Quick Check 4-2

  21. 0 3 –5 3 –9 10 = / –7 14 6 –17 0 0 0 0 0 0 0 0 2 3 no, – –0.6 6 10 1 11 Adding and Subtracting Matrices Lesson 4-2 Lesson Quiz Find each sum or difference. 1. + 2. – –3 1 4 8 –5 4 –3 –2 9 5 4 –6 2 8 0 –12 –9 6 6 –5 3. What is the additive identity for 2  4 matrices? 4. Solve the equation for x and y. 5. Are the following matrices equal? 6. Solve X – = for the matrix X. –2x –1 5 x + y 18 –3x + 4y x – 2y –16 x = –9, y = –7 = 3 0.5 – 6 2 0.50 0.4 –0.6 ; 2 5 2 3 4 3 1 5 2 7 0 6 4-2

  22. Matrix Multiplication Lesson 4-3 Check Skills You’ll Need (For help, go to Lesson 4-2.) Find each sum. 1. + + 3 5 2 8 3 5 2 8 3 5 2 8 –4 7 –4 7 –4 7 –4 7 –4 7 2. + + + + –1 3 4 0 –2 –5 –1 3 4 0 –2 –5 –1 3 4 0 –2 –5 –1 3 4 0 –2 –5 3. + + + Check Skills You’ll Need 4-3

  23. –4 7 –4 7 –4 7 –4 7 –4 7 2. + + + + = –4 + (–4) + (–4) + (–4) + (–4) 7 + 7 + 7 + 7 + 7 –20 35 = –1 3 4 0 –2 –5 –1 3 4 0 –2 –5 –1 3 4 0 –2 –5 –1 3 4 0 –2 –5 3. + + + 4(–1) 4(3) 4(4) 4(0) 4(–2) 4(–5) –4 12 16 0 –8 –20 = = Matrix Multiplication Lesson 4-3 Check Skills You’ll Need Solutions 3 5 2 8 3 5 2 8 3 5 2 8 3 + 3 + 3 5 + 5 + 5 2 + 2 + 2 8 + 8 + 8 9 15 6 24 1. + + = = 4-3

  24. Store M1 M2 M3 A $38,500 $40,000 $44,600 B $39,000 $37,800 $43,700 38500 40000 44600 39000 37800 43700 1.08 1.08(38500) 1.08(40000) 1.08(44600) 1.08(39000) 1.08(37800) 1.08(43700) = Multiply each element by 1.08. Matrix Multiplication Lesson 4-3 Additional Examples The table shows the salaries of the three managers (M1, M2, M3) in each of the two branches (A and B) of a retail clothing company. The president of the company has decided to give each manager an 8% raise. Show the new salaries in a matrix. 4-3

  25. M1 M2 M3 41580 43200 48168 42120 40824 47196 = A B Matrix Multiplication Lesson 4-3 Additional Examples (continued) The new salaries at branch A are $41,580, $43,200, and $48,168. The new salaries at branch B are $42,120, $40,824, and $47,196. Quick Check 4-3

  26. 2 –3 0 6 –5 –1 2 9 –3M + 7N = –3 + 7 –6 9 0 –18 –35 –7 14 63 = + –41 2 14 45 = Matrix Multiplication Lesson 4-3 Additional Examples Find the sum of –3M + 7N for M = and N = . 2 –3 0 6 –5 –1 2 9 Quick Check 4-3

  27. 6 9 –12 15 27 –18 30 6 –3Y + 2 = 12 18 –24 30 27 –18 30 6 –3Y + = Scalar multiplication. 27 –18 30 6 12 18 –24 30 12 18 –24 30 –3Y = – Subtract from each side. 15 –36 54 –24 –3Y = Simplify. Multiply each side by – and simplify. 15 –36 54 –24 –5 12 –18 8 1 3 Y = – = 1 3 Matrix Multiplication Lesson 4-3 Additional Examples 6 9 –12 15 27 –18 30 6 Solve the equation –3Y + 2 = . 4-3

  28. Check: 6 9 –12 15 27 –18 30 6 –3Y + 2 = –5 12 –18 8 6 9 –12 15 27 –18 30 6 –3+ 2 Substitute. 15 –36 54 –24 12 18 –24 30 27 –18 30 6 + Multiply. 27 –18 30 6 27 –18 30 6 = Simplify. Matrix Multiplication Lesson 4-3 Additional Examples (continued) Quick Check 4-3

  29. –2 5 3 –1 4 –4 2 6 = (–2)(4) + (5)(2) = 2 –2 5 3 –1 4 –4 2 6 2 = (–2)(4) + (5)(6) = 38 –2 5 3 –1 4 –4 2 6 2 38 = (3)(4) + (–1)(2) = 10 Matrix Multiplication Lesson 4-3 Additional Examples –2 5 3 –1 4 –4 2 6 Find the product of and . Multiply a11 and b11. Then multiply a12 and b21. Add the products. The result is the element in the first row and first column. Repeat with the rest of the rows and columns. 4-3

  30. –2 5 3 –1 4 –4 2 6 2 38 10 = (3)(–4) + (–1)(6) = –18 –2 5 3 –1 4 –4 2 6 2 38 10 –18 The product of and is . Matrix Multiplication Lesson 4-3 Additional Examples (continued) Quick Check 4-3

  31. 109 76 18 22 = (18)(109) + (22)(76) = 3634 Matrix Multiplication Lesson 4-3 Additional Examples Quick Check Matrix A gives the prices of shirts and jeans on sale at a discount store. Matrix B gives the number of items sold on one day. Find the income for the day from the sales of the shirts and jeans. Prices Number of Items Sold Shirts Jeans Shirts 109 Jeans 76 A = $18 $22B = Multiply each price by the number of items sold and add the products. The store’s income for the day from the sales of shirts and jeans was $3634. 4-3

  32. QP (3 4) (3 3) not equal Matrix Multiplication Lesson 4-3 Additional Examples 3 –1 2 5 9 0 0 1 8 6 5 7 0 2 0 3 1 1 –1 5 2 Use matrices P = and Q = . Determine whether products PQ and QP are defined or undefined. Find the dimensions of each product matrix. PQ (33) (34) 34 product equal matrix Product PQ is defined and is a 3  4 matrix. Product PQ is undefined, because the number of columns of Q is not equal to the number of rows in P. Quick Check 4-3

  33. –6 4 –2 –27 18 –9 –12 8 –4 16 24 –8 0 –40 32 27 –29 Matrix Multiplication Lesson 4-3 Lesson Quiz Use matrices A, B, C, and D. 2 3 –1 0 –5 4 –7 1 0 2 6 –6 A = B = C = D = 2 9 4 –3 2 –1 1. Find 8A. 2. Find AC. 3. Find CD. 4. Is BD defined or undefined? 5. What are the dimensions of (BC)D? undefined 2  3 4-3

  34. 1 3 Geometric Transformations with Matrices Lesson 4-4 Check Skills You’ll Need (For help, go to Lesson 2-6.) Without using graphing technology, graph each function and its translation. Write the new function. 1 2 1.y = x + 2; left 4 units 2.ƒ(x) = x + 2; up 5 units 3.g(x) = |x|; right 3 units 4.y = x; down 2 units 5.y = |x – 3|; down 2 units 6. ƒ(x) = –2|x|; right 2 units Check Skills You’ll Need 4-4

  35. 1 2 1 2 Geometric Transformations with Matrices Lesson 4-4 Check Skills You’ll Need 1.y = x + 2; left 4 units: y = x + 6  Solutions 2.ƒ(x) = x + 2; up 5 units: ƒ(x) = x + 7; 3.g(x) = |x| right 3 units: g(x) = |x – 3|  4.y = x; down 2 units: y = x – 2 4-4

  36. 1 3 1 3 Geometric Transformations with Matrices Lesson 4-4 Check Skills You’ll Need 5.y = |x – 3|; down 2 units: y = |x – 3| – 2 Solutions (continued) 6.ƒ(x) = –2|x| right 2 units: ƒ(x) = –2|x + 4| 4-4

  37. Vertices of Translation Vertices of the Triangle Matrix the image Subtract 3 from each x-coordinate. Add 1 to each y-coordinate. A B CA B C 1 3 2 –2 1 3 –3 –3 –3 1 1 1 –2 0 –1 –1 2 4 + = Geometric Transformations with Matrices Lesson 4-4 Additional Examples Triangle ABC has vertices A(1, –2), B(3, 1) and C(2, 3). Use a matrix to find the vertices of the image translated 3 units left and 1 unit up. Graph ABC and its image ABC. The coordinates of the vertices of the image are A (–2, –1), B (0, 2), C (–1, 4). Quick Check 4-4

  38. A B C D EABCDE 4 3 2 3 4 3 0 2 3 –1 –2 3 2 –2 –3 0 0 2 – – 2 – –2 0 2 3 = Multiply. 4 3 4 3 4 3 4 3 4 3 The new coordinates are A (0, 2),B ( , ),C (2, – ), D (– , –2), and E (– , 0). 2 3 4 3 Geometric Transformations with Matrices Lesson 4-4 Additional Examples 2 3 The figure in the diagram is to be reduced by a factor of . Find the coordinates of the vertices of the reduced figure. Write a matrix to represent the coordinates of the vertices. Quick Check 4-4

  39. 1 0 0 –1 2 3 4 –1 0 –2 2 3 4 1 0 2 = –1 0 0 1 2 3 4 –1 0 –2 –2 –3 –4 –1 0 –2 = 0 1 1 0 2 3 4 –1 0 –2 –1 0 –2 2 3 4 = 0 –1 –1 0 2 3 4 –1 0 –2 1 0 2 –2 –3 –4 = Geometric Transformations with Matrices Lesson 4-4 Additional Examples Reflect the triangle with coordinates A(2, –1), B(3, 0), and C(4, –2) in each line. Graph triangle ABC and each image on the same coordinate plane. a.x-axis b.y-axis c.y = x d.y = –x 4-4

  40. a.x-axis 1 0 0 –1 2 3 4 –1 0 –2 2 3 4 1 0 2 = b.y-axis –1 0 0 1 2 3 4 –1 0 –2 –2 –3 –4 –1 0 –2 = c.y = x 1 0 0 1 2 3 4 –1 0 –2 –1 0 –2 2 3 4 = 0 –1 –1 0 2 3 4 –1 3 –2 1 0 2 –2 –3 –4 = Geometric Transformations with Matrices Lesson 4-4 Additional Examples (continued) d.y = –x Quick Check 4-4

  41. 0 –1 1 0 2 3 4 –1 0 –2 1 0 2 2 3 4 = –1 0 0 –1 2 3 4 –1 0 –2 –2 –3 –4 1 0 2 = 0 1 –1 0 2 3 4 –1 0 –2 –1 0 –2 –2 –3 –4 = 1 0 0 1 2 3 4 –1 0 –2 2 3 4 –1 0 –2 = Geometric Transformations with Matrices Lesson 4-4 Additional Examples Rotate the triangle from Additional Example 3 as indicated. Graph the triangle ABC and each image on the same coordinate plane. a. 90 b. 180 c. 270 d. 360 Quick Check 4-4

  42. –1 2 1 1 2 –2 –7 –7 –7 3 3 3 –8 –5 –6 4 5 1 + = –1 2 1 1 2 –2 –5 10 5 5 10 –10 5 = 0 –1 –1 0 –1 2 1 1 2 –2 –1 –2 2 1 –2 –1 + = Geometric Transformations with Matrices Lesson 4-4 Lesson Quiz For these questions, use triangle ABC with vertices A(–1, 1), B(2, 2), and C(1, –2). 1. Write a matrix equation that represents a translation of triangle ABC 7 units left and 3 units up. 2. Write a matrix equation that represents a dilation of triangle ABC with a scale factor of 5. 3. Use matrix multiplication to reflect triangle ABC in the line y = –x. Then draw the preimage and image on the same coordinate plane. 4-4

  43. 2 X 2 Matrices, Determinants, and Inverses Lesson 4-5 Check Skills You’ll Need (For help, go to Skills Handbook page 845.) Simplify each group of expressions. 1a. 3(4) b. 2(6) c. 3(4) – 2(6) 2a. 3(–4) b. 2(–6) c. 3(–4) – 2(–6) 3a. –3(–4) b. 2(–6) c. –3(–4) – 2(–6) 4a. –3(4) b. –2(–6) c. –3(4) – (–2)(–6) Check Skills You’ll Need 4-5

  44. 2 X 2 Matrices, Determinants, and Inverses Lesson 4-5 Check Skills You’ll Need Solutions 1a. 3(4) = 12 1b. 2(6) = 12 1c. 3(4) – 2(6) = 12 – 12 = 0 2a. 3(–4) = –12 2b. 2(–6) = –12 2c. 3(–4) – 2(–6) = –12 – (–12) = –12 + 12 = 0 3a. –3(–4) = 12 3b. 2(–6) = –12 3c. –3(–4) – 2(–6) = 12 – (–12) = 12 + 12 = 24 4a. –3(4) = –12 4b. –2(–6) = 12 4c. –3(4) – (–2)(–6) = –12 – 12 = –24 4-5

  45. 3 –1 7 1 0.1 0.1 –0.7 0.3 AB = (3)(0.1) + (–1)(–0.7) (3)(0.1) + (–1)(0.3) (7)(0.1) + (1)(–0.7) (7)(0.1) + (1)(0.3) = 1 0 0 1 = 2 X 2 Matrices, Determinants, and Inverses Lesson 4-5 Additional Examples Show that matrices A and B are multiplicative inverses. 3 –1 7 1 0.1 0.1 –0.7 0.3 A = B = AB = I, so B is themultiplicative inverse of A. Quick Check 4-5

  46. 78 –5–9 = = (7)(–9) – (8)(–5) = –23 4–3 56 = = (4)(6) – (–3)(5) = 39 a–b ba = = (a)(a) – (–b)(b) = a2 + b2 2 X 2 Matrices, Determinants, and Inverses Lesson 4-5 Additional Examples Evaluate each determinant. 7 8 –5 –9 a. det b. det c. det 4 –3 5 6 a –b ba Quick Check 4-5

  47. = / 12 4 9 3 6 5 25 20 Since the determinant 0, the inverse of Y exists. 2 X 2 Matrices, Determinants, and Inverses Lesson 4-5 Additional Examples Determine whether each matrix has an inverse. If it does, find it. a.X = Find det X. ad – bc = (12)(3) – (4)(9) Simplify. = 0 Since det X = 0, the inverse of X does not exist. b.Y = Find det Y. ad – bc = (6)(20) – (5)(25) Simplify. = –5 4-5

  48. 20–5 Change signs. –256 Switch positions. 1 det Y 20 –5 Use the determinant to –25 6 write the inverse. 1 det Y = 20 –5 Substitute –5 for the –25 6 determinant. 1 5 = – –4 1 5 –1.2 = Multiply. 2 X 2 Matrices, Determinants, and Inverses Lesson 4-5 Additional Examples (continued) Y–1 = Quick Check 4-5

  49. d–b –ca Use the definition of inverse. 1 ad – bc A–1 = 11–25 –49 1 (9)(11) – (25)(4) Substitute. = –11 25 4 –9 Simplify. = –11 25 4 –9 3 –7 Substitute. X = 2 X 2 Matrices, Determinants, and Inverses Lesson 4-5 Additional Examples 9 25 4 11 3 –7 Solve X = for the matrix X. The matrix equation has the form AX = B. First find A–1. Use the equation X = A–1B. 4-5

  50. Multiply and simplify. (–11)(3) + (25)(–7) (4)(3) + (–9)(–7) –208 75 = = Check: Use the original equation. 9 25 4 11 3 –7 X = 9 25 4 11 –208 75 3 –7 Substitute. 9(–208) + 25(75) 4(–208) + 11(75) 3 –7 Multiply and simplify. 3 –7 3 –7 = 2 X 2 Matrices, Determinants, and Inverses Lesson 4-5 Additional Examples (continued) Quick Check 4-5