Three different models of cars are produced at three different plants: A, B, and C. The matrix F and
matrix S represent first quarter and second quarter productions, respectively, as shown below:
What is the total production by each plant for each model? (10 marks)
(4) Find the cofactors and determinants of 3x3 matrix A (10 marks)
A= [
8 2 3
1 9 4
4 3 6
]
(5) Use Crammer’s rule to compute the solutions of the following systems. (15 marks)
4x – y + 2z = 13
x + 2y - 2z = 0
-x + y + z = 5
First Quarter (F) Second Quarter (S)
Model 1 Model 2 Model 3 Model 1 Model 2 Model 3
Plant X 25 42 48 Plant X 27 44 51
Plant Y 33 40 66 Plant Y 35 39 62
Plant Z 35 48 50 Plant Z 33 50
Plant X:
Model 1: 25 (F) + 27 (S) = 52
Model 2: 42 (F) + 44 (S) = 86
Model 3: 48 (F) + 51 (S) = 99
Plant Y:
Model 1: 33 (F) + 35 (S) = 68
Model 2: 40 (F) + 39 (S) = 79
Model 3: 66 (F) + 62 (S) = 128
Plant Z:
Model 1: 35 (F) + 33 (S) = 68
Model 2: 48 (F) + 50 (S) = 98
Model 3: 50 (F) + 48 (S) = 98
Cofactors and determinants of matrix A:
Cofactor of a11 = (9(6) - 4(3)) = 42
Cofactor of a12 = -(1(6) - 4(3)) = -9
Cofactor of a13 = (1(3) - 9(4)) = -33
Determinant of A = 8(42) - 2(-9) + 3(-33) = 336 + 18 - 99 = 255
Using Cramer's Rule to solve the system of equations:
x = det(Ax) / det(A) = det([13 -1 2; 0 2 -2; 5 1 1]) / det(A) = 21 / 255 = 0.0824
y = det(Ay) / det(A) = det([4 13 2; 1 0 -2; -1 5 1]) / det(A) = -68 / 255 = -0.2667
z = det(Az) / det(A) = det([4 -1 13; 1 2 0; -1 1 5]) / det(A) = 87 / 255 = 0.3412
Therefore, the solutions to the system of equations are x = 0.0824, y = -0.2667, z = 0.3412.