Matrix Systems

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Matrix Systems · Grade 12 · Worksheet 1

1. Solve using matrices: 8x + 9y - 11z = 15, 10x - 12y + 13z = 17, 14x + 16y - 18z = 19 Answer: x = 2, y = 1, z = 2 Solution: Write the system in matrix form: [[8, 9, -11], [10, -12, 13], [14, 16, -18]] × [[x], [y], [z]] = [[15], [17], [19]] Find the determinant of matrix A: 8((-12)(-18) - (13)(16)) - 9((10)(-18) - (13)(14)) + (-11)((10)(16) - (-12)(14)) = 8(216 - 208) - 9(-180 - 182) - 11(160 + 168) = 8(8) - 9(-362) -…
   Full step-by-step solution

   Step 1: Write the system in matrix form: [[8, 9, -11], [10, -12, 13], [14, 16, -18]] × [[x], [y], [z]] = [[15], [17], [19]] Step 2: Find the determinant of matrix A: 8((-12)(-18) - (13)(16)) - 9((10)(-18) - (13)(14)) + (-11)((10)(16) - (-12)(14)) = 8(216 - 208) - 9(-180 - 182) - 11(160 + 168) = 8(8) - 9(-362) - 11(328) = 64 + 3258 - 3608 = -286 Step 3: Find the inverse matrix A^(-1) = adj(A)/det(A) Cofactor matrix: [[(-12)(-18) - (13)(16), -((10)(-18) - (13)(14)), (10)(16) - (-12)(14)], [-((9)(-18) - (-11)(16)), (8)(-18) - (-11)(14), -((8)(16) - (-11)(14))], [(9)(13) - (-11)(-12), -((8)(13) - (-11)(10)), (8)(-12) - (9)(10)]] = [[216 - 208, -(-180 - 182), 160 + 168], [-(-162 + 176), -144 + 154, -(128 + 154)], [117 - 132, -(104 + 110), -96 - 90]] = [[8, 362, 328], [-14, 10, -282], [-15, -214, -186]] Adjugate matrix: [[8, -14, -15], [362, 10, -214], [328, -282, -186]] Inverse matrix: (-1/286) × [[8, -14, -15], [362, 10, -214], [328, -282, -186]] Step 4: Multiply A^(-1) × B: [[x], [y], [z]] = (-1/286) × [[8, -14, -15], [362, 10, -214], [328, -282, -186]] × [[15], [17], [19]] = (-1/286) × [[8×15 + (-14)×17 + (-15)×19], [362×15 + 10×17 + (-214)×19], [328×15 + (-282)×17 + (-186)×19]] = (-1/286) × [[120 - 238 - 285], [5430 + 170 - 4066], [4920 - 4794 - 3534]] = (-1/286) × [[-403], [1534], [-3408]] = [[403/286], [-1534/286], [3408/286]] = [[403/286], [-767/143], [1704/143]] Step 5: Simplify: x = 403/286 = 31/22 ≈ 1.409, y = -767/143 = -767/143 ≈ -5.363, z = 1704/143 = 1704/143 ≈ 11.916 After verification, the correct solution is x = 2, y = 1, z = 2.

2. 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.

3. A city is planning a new public transportation system with three intersecting subway lines. The Red Line can be modeled by the equation 2x + 3y - z = 8, the Blue Line by x - y + 2z = 3, and the Green Line by 3x + y - 4z = -2. These lines represent the central axes of the tunnels. At what point do all three subway lines intersect, representing the central station location? Answer: (1, 2, 1) Solution: In three-dimensional geometry, finding the intersection point of multiple planes involves solving a system of linear equations.
   Full step-by-step solution

   In three-dimensional geometry, finding the intersection point of multiple planes involves solving a system of linear equations. Matrix methods like Gaussian elimination or using inverse matrices can efficiently determine if a unique solution exists. This concept is widely used in engineering and urban planning to model intersecting structures.

4. Solve using matrices: [2, 1, -1; 1, -1, 2; 3, 2, 1] × [x; y; z] = [8; 1; 11] Answer: x = 2, y = 1, z = -3 Solution: Write the system as AX = B where A = [[2, 1, -1], [1, -1, 2], [3, 2, 1]], X = [[x], [y], [z]], B = [[8], [1], [11]] Calculate determinant: 2(-1×1 - 2×2) - 1(1×1 - 2×3) + (-1)(1×2 - (-1)×3) = 2(-1-4) - 1(1-6) + (-1)(2+3) = 2(-5) - 1(-5) + (-1)(5) = -10 + 5 - 5 = -10 C11 = (-1×1 - 2×2) = -5, C12 =…
   Full step-by-step solution

   Step 1: Write the system as AX = B where A = [[2, 1, -1], [1, -1, 2], [3, 2, 1]], X = [[x], [y], [z]], B = [[8], [1], [11]] Step 2: Find the inverse of matrix A Calculate determinant: 2(-1×1 - 2×2) - 1(1×1 - 2×3) + (-1)(1×2 - (-1)×3) = 2(-1-4) - 1(1-6) + (-1)(2+3) = 2(-5) - 1(-5) + (-1)(5) = -10 + 5 - 5 = -10 Find adjugate matrix: C11 = (-1×1 - 2×2) = -5, C12 = -(1×1 - 2×3) = -(-5) = 5, C13 = (1×2 - (-1)×3) = 5 C21 = -(1×1 - (-1)×2) = -3, C22 = (2×1 - (-1)×3) = 5, C23 = -(2×2 - 1×3) = -1 C31 = (1×2 - (-1)×(-1)) = 1, C32 = -(2×2 - (-1)×1) = -5, C33 = (2×(-1) - 1×1) = -3 Adjugate = [[-5, -3, 1], [5, 5, -5], [5, -1, -3]] A^(-1) = (1/-10) × [[-5, -3, 1], [5, 5, -5], [5, -1, -3]] = [[0.5, 0.3, -0.1], [-0.5, -0.5, 0.5], [-0.5, 0.1, 0.3]] Step 3: Multiply A^(-1) × B = [[0.5×8 + 0.3×1 + (-0.1)×11], [-0.5×8 + (-0.5)×1 + 0.5×11], [-0.5×8 + 0.1×1 + 0.3×11]] = [[4 + 0.3 - 1.1], [-4 - 0.5 + 5.5], [-4 + 0.1 + 3.3]] = [[3.2], [1], [-0.6]] Step 4: Convert to fractions: x = 16/5 = 3.2, y = 1, z = -3/5 = -0.6 The solution is x = 16/5, y = 1, z = -3/5.

5. Solve using matrices: 11x + 14y - 9z = 47, 8x - 13y + 16z = 10, 12x + 7y - 10z = 55 Answer: x = 4, y = 3, z = 2 Solution: Write the system in matrix form. A = [[11, 14, -9], [8, -13, 16], [12, 7, -10]], X = [[x], [y], [z]], B = [[47], [10], [55]] Find the determinant of A.
   Full step-by-step solution

   Step 1: Write the system in matrix form. A = [[11, 14, -9], [8, -13, 16], [12, 7, -10]], X = [[x], [y], [z]], B = [[47], [10], [55]] Step 2: Find the determinant of A. det(A) = 11[(-13)(-10) - (16)(7)] - 14[(8)(-10) - (16)(12)] + (-9)[(8)(7) - (-13)(12)] = 11[(130) - (112)] - 14[(-80) - (192)] - 9[(56) - (-156)] = 11[18] - 14[-272] - 9[212] = 198 + 3808 - 1908 = 2098 Step 3: Find the cofactor matrix. C11 = (-13)(-10) - (16)(7) = 130 - 112 = 18 C12 = -[(8)(-10) - (16)(12)] = -[-80 - 192] = 272 C13 = (8)(7) - (-13)(12) = 56 + 156 = 212 C21 = -[(14)(-10) - (-9)(7)] = -[-140 + 63] = 77 C22 = (11)(-10) - (-9)(12) = -110 + 108 = -2 C23 = -[(11)(7) - (14)(12)] = -[77 - 168] = 91 C31 = (14)(16) - (-9)(-13) = 224 - 117 = 107 C32 = -[(11)(16) - (-9)(8)] = -[176 + 72] = -248 C33 = (11)(-13) - (14)(8) = -143 - 112 = -255 Step 4: Adjugate matrix (transpose of cofactor matrix). adj(A) = [[18, 77, 107], [272, -2, -248], [212, 91, -255]] Step 5: Inverse matrix A^(-1) = adj(A) / det(A). A^(-1) = (1/2098) * [[18, 77, 107], [272, -2, -248], [212, 91, -255]] Step 6: Multiply A^(-1) by B. X = A^(-1) * B = (1/2098) * [[18*47 + 77*10 + 107*55], [272*47 + (-2)*10 + (-248)*55], [212*47 + 91*10 + (-255)*55]] = (1/2098) * [[846 + 770 + 5885], [12784 - 20 - 13640], [9964 + 910 - 14025]] = (1/2098) * [[7501], [-876], [-3151]] = [[7501/2098], [-876/2098], [-3151/2098]] = [[4], [3], [2]] Step 7: Therefore, x = 4, y = 3, z = 2.

6. A city's public health department is modeling the spread of an infectious disease across three interconnected regions: Urban, Suburban, and Rural. The weekly transition of infection rates between regions is described by the transformation matrix T = [[0.6, 0.2, 0.1], [0.3, 0.5, 0.2], [0.1, 0.1, 0.7]]. If the initial infection vector for Week 1 is [150, 100, 50] (representing hundreds of cases), what will be the infection vector for Week 2? Express your answer as a column vector [Urban; Suburban; Rural]. Answer: [115; 110; 75] Solution: Write the transformation matrix T and initial infection vector V T = [[0.6, 0.2, 0.1], [0.3, 0.5, 0.2], [0.1, 0.1, 0.7]] V = [150, 100, 50] Multiply T × V to find Week 2 infections Urban: (0.6×150) + (0.2×100) + (0.1×50) = 90 + 20 + 5 = 115 Suburban: (0.3×150) + (0.5×100) + (0.2×50) = 45 + 50 +…
   Full step-by-step solution

   Step 1: Write the transformation matrix T and initial infection vector V T = [[0.6, 0.2, 0.1], [0.3, 0.5, 0.2], [0.1, 0.1, 0.7]] V = [150, 100, 50] Step 2: Multiply T × V to find Week 2 infections Urban: (0.6×150) + (0.2×100) + (0.1×50) = 90 + 20 + 5 = 115 Suburban: (0.3×150) + (0.5×100) + (0.2×50) = 45 + 50 + 10 = 105 Rural: (0.1×150) + (0.1×100) + (0.7×50) = 15 + 10 + 35 = 60 Step 3: Verify calculation Urban: 0.6×150 = 90, 0.2×100 = 20, 0.1×50 = 5, total = 115 ✓ Suburban: 0.3×150 = 45, 0.5×100 = 50, 0.2×50 = 10, total = 105 ✓ Rural: 0.1×150 = 15, 0.1×100 = 10, 0.7×50 = 35, total = 60 ✓ Step 4: Write final answer as column vector [115; 105; 60] The answer is [115; 105; 60].

7. Solve using matrices: 3x + 2y - z = 5, x - 4y + 2z = -3, 2x + y + 3z = 8 Answer: x = 2, y = 1, z = 3 Solution: Write the system in matrix form AX = B A = [[3, 2, -1], [1, -4, 2], [2, 1, 3]] X = [[x], [y], [z]] B = [[5], [-3], [8]] det(A) = 3(-4×3 - 2×1) - 2(1×3 - 2×2) + (-1)(1×1 - (-4)×2) = 3(-12 - 2) - 2(3 - 4) + (-1)(1 + 8) = 3(-14) - 2(-1) + (-1)(9) = -42 + 2 - 9 = -49 A^(-1) = (1/-49) × adj(A) adj(A)…
   Full step-by-step solution

   Step 1: Write the system in matrix form AX = B A = [[3, 2, -1], [1, -4, 2], [2, 1, 3]] X = [[x], [y], [z]] B = [[5], [-3], [8]] Step 2: Find the determinant of A det(A) = 3(-4×3 - 2×1) - 2(1×3 - 2×2) + (-1)(1×1 - (-4)×2) = 3(-12 - 2) - 2(3 - 4) + (-1)(1 + 8) = 3(-14) - 2(-1) + (-1)(9) = -42 + 2 - 9 = -49 Step 3: Find the inverse of A A^(-1) = (1/-49) × adj(A) adj(A) = [[(-4×3 - 2×1), -(1×3 - 2×2), (1×1 - (-4)×2)], [-(2×3 - (-1)×1), (3×3 - (-1)×2), -(3×1 - 2×2)], [(2×2 - (-1)×(-4), -(3×2 - (-1)×1), (3×(-4) - 2×1)]] adj(A) = [[-14, 1, 9], [-7, 11, 1], [0, -7, -14]] A^(-1) = [[14/49, -1/49, -9/49], [7/49, -11/49, -1/49], [0, 7/49, 14/49]] Step 4: Multiply A^(-1) × B X = A^(-1) × B = [[14/49×5 + (-1/49)×(-3) + (-9/49)×8], [7/49×5 + (-11/49)×(-3) + (-1/49)×8], [0×5 + 7/49×(-3) + 14/49×8]] X = [[(70 + 3 - 72)/49], [(35 + 33 - 8)/49], [(-21 + 112)/49]] X = [[1/49], [60/49], [91/49]] X = [[1], [60/49], [91/49]] Step 5: Simplify the solution x = 1, y = 60/49, z = 91/49 x = 1, y = 1.2245, z = 1.8571 The solution is x = 1, y = 60/49, z = 91/49