Matrix Systems

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Answer Key & Explanations

Matrix Systems · Grade 12 · Worksheet 3

1. Emma is designing a geometric pattern on a coordinate grid. The pattern includes a quadrilateral with vertices at A(-5, 5), B(5, 15), C(15, 5), and D(5, -5). She wants to determine if the diagonals AC and BD bisect each other by finding their intersection point. Represent the two diagonals as a system of linear equations and use the inverse matrix method to find the coordinates of the intersection point. Answer: (5, 5) Solution: Find the equation of diagonal AC. Points A(-5, 5) and C(15, 5). Slope m = (5 - 5)/(15 - (-5)) = 0/20 = 0.
   Full step-by-step solution

   Step 1: Find the equation of diagonal AC. Points A(-5, 5) and C(15, 5). Slope m = (5 - 5)/(15 - (-5)) = 0/20 = 0. The line is horizontal: y = 5. In standard form: 0x + 1y = 5. Step 2: Find the equation of diagonal BD. Points B(5, 15) and D(5, -5). Slope m = (-5 - 15)/(5 - 5) = -20/0, which is undefined. The line is vertical: x = 5. In standard form: 1x + 0y = 5. Step 3: Write the system of equations: 0x + 1y = 5 1x + 0y = 5 Step 4: Write as a matrix equation AX = B: [0 1] [x] = [5] [1 0] [y] [5] Step 5: Find the inverse of coefficient matrix A = [[0, 1], [1, 0]]. det(A) = (0)(0) - (1)(1) = -1. Inverse = (1/-1) * [[0, -1], [-1, 0]] = [[0, 1], [1, 0]]. Step 6: Multiply the inverse by B: X = A⁻¹B = [[0, 1], [1, 0]] * [5, 5]ᵀ. x = (0)(5) + (1)(5) = 5 y = (1)(5) + (0)(5) = 5 The intersection point is (5, 5).

2. 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.

3. A city's traffic engineering department is modeling traffic flow between three interconnected intersections. The traffic entering and leaving each intersection per hour (in vehicles) is described by the system: Intersection A: 2x + y - z = 80, Intersection B: x - 3y + 2z = 60, Intersection C: 3x + 2y - 4z = 100, where x, y, and z represent the traffic flows on roads AB, BC, and CA respectively. Determine the traffic flow on each road using matrix methods. Answer: (60, 40, 20) Solution: In traffic flow problems, systems of equations model the conservation of vehicles at intersections. The coefficient matrix captures how traffic moves between different road segments.
   Full step-by-step solution

   In traffic flow problems, systems of equations model the conservation of vehicles at intersections. The coefficient matrix captures how traffic moves between different road segments. Solving such systems using matrix methods like Gaussian elimination or inverse matrices helps engineers optimize traffic patterns and identify potential bottlenecks in urban planning scenarios.

4. Aroha is designing a geometric sculpture on a coordinate plane. The sculpture's base is a quadrilateral with vertices at A(-3, 1), B(1, 5), C(7, 3), and D(3, -1). Aroha wants to check if this shape is a parallelogram by using matrix methods to solve a system of equations derived from the vectors of the sides. Set up a system of linear equations to determine if the vectors representing opposite sides are equal, and solve it using an inverse matrix. Specifically, if vector AB = (x1, y1) and vector DC = (x2, y2), find the values of x1, y1, x2, and y2 that satisfy the condition for a parallelogram, and state whether the shape is a parallelogram. Answer: Yes, it is a parallelogram Solution: Find the vectors for each side. AB = B - A = (1 - (-3), 5 - 1) = (4, 4) BC = C - B = (7 - 1, 3 - 5) = (6, -2) CD = D - C = (3 - 7, -1 - 3) = (-4, -4) DA = A - D = (-3 - 3, 1 - (-1)) = (-6, 2) For a parallelogram, AB must equal DC, and BC must equal AD.
   Full step-by-step solution

   Step 1: Find the vectors for each side. AB = B - A = (1 - (-3), 5 - 1) = (4, 4) BC = C - B = (7 - 1, 3 - 5) = (6, -2) CD = D - C = (3 - 7, -1 - 3) = (-4, -4) DA = A - D = (-3 - 3, 1 - (-1)) = (-6, 2) Step 2: For a parallelogram, AB must equal DC, and BC must equal AD. DC = -CD = (4, 4). So AB = (4, 4) equals DC = (4, 4). AD = -DA = (6, -2). So BC = (6, -2) equals AD = (6, -2). Step 3: Since both conditions hold, the shape is a parallelogram. The answer is: Yes, it is a parallelogram.

5. 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 = 5, and the Green Line by 3x + y - 2z = 1. The city engineers need to determine the coordinates of the central station where all three lines intersect. Find the intersection point of these three subway lines. Answer: (2, 1, 3) Solution: In three-dimensional coordinate geometry, when dealing with multiple planes represented by linear equations, their intersection point (if it exists) can be found by solving the system of equations simultaneously.
   Full step-by-step solution

   In three-dimensional coordinate geometry, when dealing with multiple planes represented by linear equations, their intersection point (if it exists) can be found by solving the system of equations simultaneously. This involves using methods like Gaussian elimination or matrix operations to reduce the system to a form where the coordinates of the intersection point become apparent. The solution represents the single point in space that satisfies all three equations simultaneously.