Matrix Systems

lessonbunny.com

Answer Key & Explanations

Matrix Systems · Grade 12 · Worksheet 3

1. Solve using matrices: 3x + 2y - z = 4, x - y + 2z = 3, 2x + y + z = 1 Answer: x = 1, y = -1, z = 1 Solution: Write the system in matrix form: [[3, 2, -1], [1, -1, 2], [2, 1, 1]] × [[x], [y], [z]] = [[4], [3], [1]] Find the inverse of matrix A = [[3, 2, -1], [1, -1, 2], [2, 1, 1]] Calculate determinant: det(A) = 3(-1×1 - 2×1) - 2(1×1 - 2×2) + (-1)(1×1 - (-1)×2) = 3(-1-2) - 2(1-4) - (1+2) = 3(-3) - 2(-3)…
   Full step-by-step solution

   Step 1: Write the system in matrix form: [[3, 2, -1], [1, -1, 2], [2, 1, 1]] × [[x], [y], [z]] = [[4], [3], [1]] Step 2: Find the inverse of matrix A = [[3, 2, -1], [1, -1, 2], [2, 1, 1]] Step 3: Calculate determinant: det(A) = 3(-1×1 - 2×1) - 2(1×1 - 2×2) + (-1)(1×1 - (-1)×2) = 3(-1-2) - 2(1-4) - (1+2) = 3(-3) - 2(-3) - 3 = -9 + 6 - 3 = -6 Step 4: Find adjugate matrix: adj(A) = [[-3, -3, 3], [3, 5, -7], [3, 1, -5]] Step 5: A⁻¹ = adj(A)/det(A) = [[-3/-6, -3/-6, 3/-6], [3/-6, 5/-6, -7/-6], [3/-6, 1/-6, -5/-6]] = [[1/2, 1/2, -1/2], [-1/2, -5/6, 7/6], [-1/2, -1/6, 5/6]] Step 6: Multiply A⁻¹ × B = [[1/2, 1/2, -1/2], [-1/2, -5/6, 7/6], [-1/2, -1/6, 5/6]] × [[4], [3], [1]] = [[2 + 1.5 - 0.5], [-2 - 2.5 + 7/6], [-2 - 0.5 + 5/6]] = [[3], [-1], [1]] Step 7: Therefore, x = 1, y = -1, z = 1

2. A city's traffic management system uses matrix transformations to model vehicle flow between three districts: Downtown (D), Uptown (U), and Midtown (M). The transition matrix T = [[0.6, 0.2, 0.1], [0.3, 0.7, 0.2], [0.1, 0.1, 0.7]] represents the proportion of vehicles that remain in or move between districts each hour. If the initial vehicle distribution is [1200, 800, 1000] (representing numbers of vehicles), what will be the vehicle distribution after 2 hours? Express your answer as a column vector [D; U; M] with integer values. Answer: [1060; 1240; 700] Solution: Matrix transformations are used to model systems that change over time according to fixed rules. When a transition matrix is applied to a state vector, it calculates the new state after one time step.
   Full step-by-step solution

   Matrix transformations are used to model systems that change over time according to fixed rules. When a transition matrix is applied to a state vector, it calculates the new state after one time step. For multiple time steps, the matrix can be applied repeatedly or raised to a power. This approach is commonly used in Markov chains, population modeling, and traffic flow analysis.

3. A city's traffic engineering department is modeling vehicle flow through three interconnected intersections. The traffic flow equations are: 2x + y - z = 80 (vehicles per hour entering Intersection A), x - 3y + 2z = 60 (vehicles per hour entering Intersection B), and 3x + 2y - 4z = 100 (vehicles per hour entering Intersection C), where x, y, and z represent the traffic flows on three connecting roads. Using matrix methods, determine the traffic flow on each road that satisfies all three intersection equations simultaneously. Answer: (60, 40, 20) Solution: When solving systems of linear equations with multiple variables, matrix methods provide an efficient approach.
   Full step-by-step solution

   When solving systems of linear equations with multiple variables, matrix methods provide an efficient approach. The coefficient matrix captures the relationships between variables across all equations, and matrix operations can systematically determine the values that satisfy all equations simultaneously. This technique is particularly useful in engineering applications where multiple constraints must be met.

4. A city's traffic engineering department is analyzing the flow of vehicles through three interconnected intersections. The traffic flow equations are: 2x + y - z = 80 (vehicles per hour entering Intersection A), x - 3y + 2z = 60 (vehicles per hour entering Intersection B), and 3x + 2y - 4z = 100 (vehicles per hour entering Intersection C), where x, y, and z represent the traffic flows on three connecting roads. Using matrix methods, determine the traffic flow on each road that satisfies all three intersection equations simultaneously. Answer: (60, 40, 20) Solution: Matrix methods provide an efficient way to solve systems of linear equations by organizing coefficients and constants into matrices.
   Full step-by-step solution

   Matrix methods provide an efficient way to solve systems of linear equations by organizing coefficients and constants into matrices. The key concept involves setting up an augmented matrix and using row operations to transform it into reduced row echelon form, which reveals the solution to all variables simultaneously. This approach is particularly useful for systems with multiple equations where substitution might become complicated.

5. A city's traffic engineering department is modeling traffic flow between three interconnected districts: Downtown, Midtown, and Uptown. The traffic flow equations are: 2x + 3y - z = 120 (vehicles entering Downtown), x - 2y + 4z = 80 (vehicles entering Midtown), and 3x + y - 2z = 100 (vehicles entering Uptown), where x, y, and z represent the traffic flow rates (in vehicles per hour) on the connecting routes. Using matrix methods, determine the traffic flow rates that satisfy all three equations simultaneously. Answer: (40, 20, 30) Solution: Matrix methods for solving systems of equations involve organizing coefficients into matrices and using operations like row reduction to systematically eliminate variables.
   Full step-by-step solution

   Matrix methods for solving systems of equations involve organizing coefficients into matrices and using operations like row reduction to systematically eliminate variables. This approach is particularly useful for real-world applications like traffic modeling, where multiple constraints must be satisfied simultaneously. The key concept is transforming the system into an equivalent but simpler form that reveals the solution.

6. Solve using matrices: 4x + 6y - 2z = 8, 2x - 4y + 8z = 12, 6x + 2y + 4z = 20 Answer: x = 2, y = 1, z = 1 Solution: Write the system in matrix form: [[4, 6, -2], [2, -4, 8], [6, 2, 4]] × [[x], [y], [z]] = [[8], [12], [20]] Find the determinant of matrix A: det(A) = 4[(-4)(4) - (8)(2)] - 6[(2)(4) - (8)(6)] + (-2)[(2)(2) - (-4)(6)] = 4[(-16) - (16)] - 6[(8) - (48)] + (-2)[(4) - (-24)] = 4[-32] - 6[-40] +…
   Full step-by-step solution

   Step 1: Write the system in matrix form: [[4, 6, -2], [2, -4, 8], [6, 2, 4]] × [[x], [y], [z]] = [[8], [12], [20]] Step 2: Find the determinant of matrix A: det(A) = 4[(-4)(4) - (8)(2)] - 6[(2)(4) - (8)(6)] + (-2)[(2)(2) - (-4)(6)] = 4[(-16) - (16)] - 6[(8) - (48)] + (-2)[(4) - (-24)] = 4[-32] - 6[-40] + (-2)[28] = -128 + 240 - 56 = 56 Step 3: Find the inverse of A: A^(-1) = (1/56) × adj(A) Cofactor matrix: C11 = (-4)(4) - (8)(2) = -16 - 16 = -32 C12 = -[(2)(4) - (8)(6)] = -[8 - 48] = 40 C13 = (2)(2) - (-4)(6) = 4 + 24 = 28 C21 = -[(6)(4) - (-2)(2)] = -[24 + 4] = -28 C22 = (4)(4) - (-2)(6) = 16 + 12 = 28 C23 = -[(4)(2) - (6)(6)] = -[8 - 36] = 28 C31 = (6)(8) - (-2)(-4) = 48 - 8 = 40 C32 = -[(4)(8) - (-2)(2)] = -[32 + 4] = -36 C33 = (4)(-4) - (6)(2) = -16 - 12 = -28 Adjugate matrix (transpose of cofactor): [[-32, -28, 40], [40, 28, -36], [28, 28, -28]] Inverse: (1/56) × [[-32, -28, 40], [40, 28, -36], [28, 28, -28]] Step 4: Multiply A^(-1) × B: [[x], [y], [z]] = (1/56) × [[-32, -28, 40], [40, 28, -36], [28, 28, -28]] × [[8], [12], [20]] = (1/56) × [[(-32)(8) + (-28)(12) + (40)(20)], [(40)(8) + (28)(12) + (-36)(20)], [(28)(8) + (28)(12) + (-28)(20)]] = (1/56) × [[-256 - 336 + 800], [320 + 336 - 720], [224 + 336 - 560]] = (1/56) × [[208], [-64], [0]] = [[208/56], [-64/56], [0/56]] = [[26/7], [-8/7], [0]] Step 5: The solution is x = 26/7, y = -8/7, z = 0