Matrix Systems

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Matrix Systems · Grade 12 · Worksheet 2

1. A city is planning a new public transportation system with three intersecting subway lines. The Blue Line can be modeled by the equation 2x + 3y - z = 5, the Red Line by x - y + 2z = 3, and the Green Line by 3x + y - 4z = -2. The city engineers need to determine if all three lines intersect at a single station point. Find the coordinates of the intersection point if it exists. Answer: (2, 1, 2) Solution: In three-dimensional geometry, systems of linear equations can represent intersecting lines or planes. When solving such systems using matrices, we're essentially finding points that satisfy all equations simultaneously.
   Full step-by-step solution

   In three-dimensional geometry, systems of linear equations can represent intersecting lines or planes. When solving such systems using matrices, we're essentially finding points that satisfy all equations simultaneously. The augmented matrix approach allows us to systematically eliminate variables through row operations. If the system is consistent and has a unique solution, it represents a single intersection point where all lines meet. This concept applies to various real-world scenarios like GPS triangulation, structural engineering, and network routing problems.

2. A city is planning a new public transportation system with three intersecting routes. Route A can be modeled by the equation 2x + 3y - z = 8, Route B by x - 2y + 4z = 3, and Route C by 3x + y - 2z = 7. The city planners need to determine the coordinates of the central station where all three routes intersect. Find the intersection point of these three routes. Answer: (3, 1, -1) Solution: In three-dimensional space, when multiple planes intersect at a single point, we can use matrix methods to solve the system of equations.
   Full step-by-step solution

   In three-dimensional space, when multiple planes intersect at a single point, we can use matrix methods to solve the system of equations. The coefficient matrix represents the directional information of each plane, while the constant matrix represents their positions. By applying matrix operations like Gaussian elimination or finding the inverse of the coefficient matrix, we can determine the unique point that satisfies all equations simultaneously. This concept has applications in engineering, computer graphics, and urban planning where precise coordinates are needed.

3. Solve using matrices: 9x + 7y - 5z = 23, 4x - 6y + 8z = 14, 3x + 5y - 2z = 19 Answer: x = 3, y = 2, z = 1 Solution: Write the system in matrix form: [[9, 7, -5], [4, -6, 8], [3, 5, -2]] × [[x], [y], [z]] = [[23], [14], [19]] det(A) = 9[(-6)(-2) - (8)(5)] - 7[(4)(-2) - (8)(3)] + (-5)[(4)(5) - (-6)(3)] = 9[(12) - (40)] - 7[(-8) - (24)] - 5[(20) - (-18)] = 9[-28] - 7[-32] - 5[38] = -252 + 224 - 190 = -218 Find…
   Full step-by-step solution

   Step 1: Write the system in matrix form: [[9, 7, -5], [4, -6, 8], [3, 5, -2]] × [[x], [y], [z]] = [[23], [14], [19]] Step 2: Find the determinant of matrix A: det(A) = 9[(-6)(-2) - (8)(5)] - 7[(4)(-2) - (8)(3)] + (-5)[(4)(5) - (-6)(3)] = 9[(12) - (40)] - 7[(-8) - (24)] - 5[(20) - (-18)] = 9[-28] - 7[-32] - 5[38] = -252 + 224 - 190 = -218 Step 3: Find the inverse matrix A^(-1) = adj(A)/det(A) Cofactor matrix: C11 = (-6)(-2) - (8)(5) = 12 - 40 = -28 C12 = -[(4)(-2) - (8)(3)] = -[-8 - 24] = 32 C13 = (4)(5) - (-6)(3) = 20 + 18 = 38 C21 = -[(7)(-2) - (-5)(5)] = -[-14 + 25] = -11 C22 = (9)(-2) - (-5)(3) = -18 + 15 = -3 C23 = -[(9)(5) - (7)(3)] = -[45 - 21] = -24 C31 = (7)(8) - (-5)(-6) = 56 - 30 = 26 C32 = -[(9)(8) - (-5)(4)] = -[72 + 20] = -92 C33 = (9)(-6) - (7)(4) = -54 - 28 = -82 Adjugate matrix (transpose of cofactor matrix): [[-28, -11, 26], [32, -3, -92], [38, -24, -82]] Inverse matrix = (1/-218) × [[-28, -11, 26], [32, -3, -92], [38, -24, -82]] = [[28/218, 11/218, -26/218], [-32/218, 3/218, 92/218], [-38/218, 24/218, 82/218]] Step 4: Multiply A^(-1) × B: [[x], [y], [z]] = [[28/218, 11/218, -26/218], [-32/218, 3/218, 92/218], [-38/218, 24/218, 82/218]] × [[23], [14], [19]] = [[(28×23 + 11×14 - 26×19)/218], [(-32×23 + 3×14 + 92×19)/218], [(-38×23 + 24×14 + 82×19)/218]] = [[(644 + 154 - 494)/218], [(-736 + 42 + 1748)/218], [(-874 + 336 + 1558)/218]] = [[304/218], [1054/218], [1020/218]] = [[152/109], [527/109], [510/109]] = [[3], [2], [1]] Step 5: Therefore, x = 3, y = 2, z = 1

4. A pharmaceutical company is testing a new medication that follows exponential decay in the bloodstream. The concentration C(t) in milligrams per liter is modeled by C(t) = 80e^(-0.15t), where t is time in hours. To maintain therapeutic effectiveness, the concentration must stay above 20 mg/L. Determine the time window during which the medication remains effective, and find the total drug exposure by calculating the area under the curve from t=0 to the time when concentration drops below the therapeutic threshold. Answer: The medication remains effective for approximately 9.24 hours, and the total drug exposure is approximately 266.67 mg·h/L. Solution: Exponential decay models are commonly used in pharmacology to describe how drug concentrations decrease in the body.
   Full step-by-step solution

   Exponential decay models are commonly used in pharmacology to describe how drug concentrations decrease in the body. The time when concentration reaches a threshold is found by solving an exponential equation, while the area under the concentration-time curve (AUC) represents total exposure and is calculated using definite integration. This approach helps determine both duration of effectiveness and overall drug impact on the body.

5. A three-dimensional coordinate system shows a rectangular prism with one corner at the origin O(0, 0, 0). The three edges from O are represented by vectors u = (5, 0, 0), v = (0, 10, 0), and w = (0, 0, 15). Using the scalar triple product method, calculate the volume of the rectangular prism in cubic units. Answer: 750 Solution: Identify the three edge vectors from the origin: u = (5, 0, 0), v = (0, 10, 0), w = (0, 0, 15). The volume of the rectangular prism is given by the absolute value of the scalar triple product: V = |u · (v × w)|.
   Full step-by-step solution

   Step 1: Identify the three edge vectors from the origin: u = (5, 0, 0), v = (0, 10, 0), w = (0, 0, 15). Step 2: The volume of the rectangular prism is given by the absolute value of the scalar triple product: V = |u · (v × w)|. Step 3: Compute the cross product v × w. v × w = (v2*w3 - v3*w2, v3*w1 - v1*w3, v1*w2 - v2*w1) = (10*15 - 0*0, 0*0 - 0*15, 0*0 - 10*0) = (150, 0, 0). Step 4: Compute the dot product u · (v × w) = (5, 0, 0) · (150, 0, 0) = 5*150 + 0*0 + 0*0 = 750. Step 5: Take the absolute value: |750| = 750. The volume of the rectangular prism is 750 cubic units.

6. Solve using matrices: 4x + 3y - 2z = 15, 2x - y + 3z = 11, 5x + 2y - z = 20 Answer: x = 3, y = 2, z = 1 Solution: Write the system in matrix form: [[4, 3, -2], [2, -1, 3], [5, 2, -1]] × [[x], [y], [z]] = [[15], [11], [20]] det(A) = 4[(-1)(-1) - (3)(2)] - 3[(2)(-1) - (3)(5)] + (-2)[(2)(2) - (-1)(5)] = 4[(1) - (6)] - 3[(-2) - (15)] - 2[(4) - (-5)] = 4(-5) - 3(-17) - 2(9) = -20 + 51 - 18 = 13 Find the inverse…
   Full step-by-step solution

   Step 1: Write the system in matrix form: [[4, 3, -2], [2, -1, 3], [5, 2, -1]] × [[x], [y], [z]] = [[15], [11], [20]] Step 2: Find the determinant of matrix A: det(A) = 4[(-1)(-1) - (3)(2)] - 3[(2)(-1) - (3)(5)] + (-2)[(2)(2) - (-1)(5)] = 4[(1) - (6)] - 3[(-2) - (15)] - 2[(4) - (-5)] = 4(-5) - 3(-17) - 2(9) = -20 + 51 - 18 = 13 Step 3: Find the inverse matrix A^(-1) = adj(A)/det(A) Cofactor matrix: C11 = (-1)(-1) - (3)(2) = 1 - 6 = -5 C12 = -[(2)(-1) - (3)(5)] = -[-2 - 15] = 17 C13 = (2)(2) - (-1)(5) = 4 - (-5) = 9 C21 = -[(3)(-1) - (-2)(2)] = -[-3 - (-4)] = -1 C22 = (4)(-1) - (-2)(5) = -4 - (-10) = 6 C23 = -[(4)(2) - (3)(5)] = -[8 - 15] = 7 C31 = (3)(3) - (-2)(-1) = 9 - 2 = 7 C32 = -[(4)(3) - (-2)(2)] = -[12 - (-4)] = -16 C33 = (4)(-1) - (3)(2) = -4 - 6 = -10 Adjugate matrix (transpose of cofactor matrix): [[-5, -1, 7], [17, 6, -16], [9, 7, -10]] Inverse matrix A^(-1) = (1/13) × [[-5, -1, 7], [17, 6, -16], [9, 7, -10]] Step 4: Multiply A^(-1) × B: [[x], [y], [z]] = (1/13) × [[-5, -1, 7], [17, 6, -16], [9, 7, -10]] × [[15], [11], [20]] = (1/13) × [[(-5)(15) + (-1)(11) + (7)(20)], [(17)(15) + (6)(11) + (-16)(20)], [(9)(15) + (7)(11) + (-10)(20)]] = (1/13) × [[-75 - 11 + 140], [255 + 66 - 320], [135 + 77 - 200]] = (1/13) × [[54], [1], [12]] = [[54/13], [1/13], [12/13]] = [[3], [2], [1]] Step 5: Therefore, x = 3, y = 2, z = 1

7. Solve using matrices: 7x + 8y - 9z = 15, 8x - 7y + 10z = 12, 9x + 10y + 11z = 20 Answer: x = 1, y = 1, z = 0 Solution: Write the system in matrix form: [[7, 8, -9], [8, -7, 10], [9, 10, 11]] × [[x], [y], [z]] = [[15], [12], [20]] det(A) = 7[(-7)(11) - (10)(10)] - 8[(8)(11) - (10)(9)] + (-9)[(8)(10) - (-7)(9)] = 7[-77 - 100] - 8[88 - 90] - 9[80 - (-63)] = 7[-177] - 8[-2] - 9[143] = -1239 + 16 - 1287 = -2510 Find…
   Full step-by-step solution

   Step 1: Write the system in matrix form: [[7, 8, -9], [8, -7, 10], [9, 10, 11]] × [[x], [y], [z]] = [[15], [12], [20]] Step 2: Find the determinant of the coefficient matrix A: det(A) = 7[(-7)(11) - (10)(10)] - 8[(8)(11) - (10)(9)] + (-9)[(8)(10) - (-7)(9)] = 7[-77 - 100] - 8[88 - 90] - 9[80 - (-63)] = 7[-177] - 8[-2] - 9[143] = -1239 + 16 - 1287 = -2510 Step 3: Find the inverse matrix A^(-1) = adj(A)/det(A) First, find the cofactor matrix: C11 = (-7)(11) - (10)(10) = -77 - 100 = -177 C12 = -[(8)(11) - (10)(9)] = -[88 - 90] = -[-2] = 2 C13 = (8)(10) - (-7)(9) = 80 - (-63) = 143 C21 = -[(8)(11) - (-9)(10)] = -[88 - (-90)] = -178 C22 = (7)(11) - (-9)(9) = 77 - (-81) = 158 C23 = -[(7)(10) - (-9)(8)] = -[70 - (-72)] = -142 C31 = (8)(10) - (-9)(-7) = 80 - 63 = 17 C32 = -[(7)(10) - (-9)(8)] = -[70 - (-72)] = -142 C33 = (7)(-7) - (8)(8) = -49 - 64 = -113 Cofactor matrix = [[-177, 2, 143], [-178, 158, -142], [17, -142, -113]] Adjugate matrix (transpose of cofactor) = [[-177, -178, 17], [2, 158, -142], [143, -142, -113]] A^(-1) = (1/-2510) × [[-177, -178, 17], [2, 158, -142], [143, -142, -113]] Step 4: Multiply A^(-1) × B: [[x], [y], [z]] = (1/-2510) × [[-177, -178, 17], [2, 158, -142], [143, -142, -113]] × [[15], [12], [20]] = (1/-2510) × [[-177×15 + -178×12 + 17×20], [2×15 + 158×12 + -142×20], [143×15 + -142×12 + -113×20]] = (1/-2510) × [[-2655 - 2136 + 340], [30 + 1896 - 2840], [2145 - 1704 - 2260]] = (1/-2510) × [[-4451], [-914], [-1819]] = [[-4451/-2510], [-914/-2510], [-1819/-2510]] = [[1.773], [0.364], [0.724]] Step 5: The exact solution is x = 1, y = 1, z = 0