Matrices #5 - Questions and Answers

Question 1 A motorcycle is traveling at 42 miles per hour on tires that measure 1.2 feet in diameter. Find the approximate angular speed of the tires in radians per minute. Answer Use the formula relating the linear speed v and angular speed w: v = wr where r is the radius. Convert 42 miles per hour to feet per minute: v = (42 miles / hr) * (1 hr / 60 min) * (5280 ft / 1 mile) = 3696 ft/min Substitute v = 3696 ft/min and r = 1.2/2 = 0.6 then find w: 3696 = ω(0.6) ω = 3696 / 0.6 ω = 6160 radians per minute Question 2 A matrix is defined as a rectangular array of numbers. Many of the arrays that you have worked have additional features, such as labels on the rows and columns or entries that are numbers with units. An array with additional features is technically not a matrix, instead we would call it a table. The difference between a matrix and a table is not vitally important in this unit, so we have worked with the underlying array of numbers and not worried about this technical detail. In more advanced courses in mathematics, like Linear Algebra, it will be important to be very precise about matrices. For now, describe and give examples of some of the ways a table can be different from a matrix. In your description and examples, be sure to include differences related to at least these three characteristics: entries, labels, and operations. Answer In a table, we can have labels, but in the matrix representation, we do not have labels. We cannot perform operations on tables, but in matrices, we can perform the mathematical operations. We can perform the mathematical operations on matrices. Question 3 Write out the system of equations that corresponds to each of the following augmented matrices: 3 2 | 8 (a) [1 5 | 7] 5 -2 1 | 3 (b) [ 2 3 -4 | 0] -2 1 4 | -1 (c) 4 -2 3 | 4] 5 2 6 | -1] 4 -3 1 2 | 4 (d) 3 1 -5 6 | 5 1 1 2 4 | 8 5 1 3 -2 | 7 Answer Step 1 Write out the system of equation to each of the following augmented matrices. Step 2 (a) [3 2 | 8] [1 5 | 7] In an augmented matrix, each row represents one equation of the system and each column represents a constant term or variable. When the columns represents the variables x₁ and x₂ Hence, the system of equations of the given augmented matrix is 3x₁ + 2x₂ = 8 x₁ + 5x₂ = 7 Step 3 (b) [5 -2 1 | 3] [2 3 -4 | 0] In an augmented matrix, each row represents one equation of the system and each column represents a constant term or variable. When the columns represents the variables x₁, x₂, and x₃ Hence, the system of equations of the given augmented matrix is 5x₁ - 2x₂ + x₃ = 3 2x₁ + 3x₂ - 4x₃ = 0 Step 4 (c) [ 2 1 4 | -1] [ 4 -2 3 | 4] [ 5 2 6 | -1] In an augmented matrix, each row represents one equation of the system and each column represents a constant term or variable. When the columns represents the variables x₁, x₂, and x₃ Hence, the system of equations of the given augmented matrix is 2x₁ + x₂ + 4x₃ = -1 4x₁ - 2x₂ + 3x₃ = 4 5x₁ + 2x₂ + 6x₃ = -1 Step 5 (d) [ 4 -3 1 2 | 4] [ 3 1 -5 6 | 5] [ 1 1 2 4 | 8] [ 5 1 3 -2 | 7] In an augmented matrix, each row represents one equation of the system and each column represents a constant term or variable. When the columns represents the variables x₁, x₂, x₃ and x₄. Hence, the system of equations of the given augmented matrix is 4x₁ - 3x₂ + x₃ + 2x₄ = 4 3x₁ + x₂ - 5x₃ + 6x₄ = 5 x₁ + x₂ + 2x₃ + 4x₄ = 8 5x₁ + x₂ + 3x₃ - 2x₄ = 7 Question 4 The length of a rectangle is 55 meters less than 33 times the width. The perimeter is 14 meters. Find the width. Answer I think that you mean 55 and 33. Start by turning the words into equations. l = length, w = width length = 33*width - 55 → l = 33w - 55 P = perimeter = 2(l+w) = 2l + 2w, which is a known equation for a rectangle. P = 14 is given: 2l + 2w = 14 Substituting what we know l equals: 2(33w - 55) + 2w = 14 Multiply by 2 to remove the parenthesis: 66w - 110 + 2w = 14 Collect terms: 68w - 110 = 14 Add 110 to both sides: 68w = 124 Divide both sides by 68: w = 124 / 68 ≈ 1.8235 Looking back, we see: l = 33w - 55 = 33(1.8235) - 55 ≈ 5.20 Always check your work. In this case, if the width ≈ 1.82 and the length is ≈ 5.2, is the perimeter ≈ 14? 2(1.8235) + 2(5.2) ≈ 3.647 + 10.4 ≈ 14.047 Remember to state the answer. The length ≈ 5.2 and the width ≈ 1.8. Question 5 Describe how to subtract matrices. Answer Step 1 In this question, we need to explain the process for subtracting matrices. Step 2 Explanation: Let A and B be two matrices of the same order, then A - B can be obtained by subtracting corresponding entries of A and B i.e. if [ a b ] [ e f ] A = [ c d ] and B= [ g h ] Then [ a - e b - f ] A - B = [ c - g d - h ] Example: Let A = [4 1] and B = [-1 8] [-6 5] [ 4 5 ] then A - B = [ 4 - (-1) 1 - 8 ] = [ 5 -7 ] [-6 -4 5 - 5] [ -10 0 ] A - B = [ 5 -7] [-10 0]