Question

Each question 25 points. Please show justification for all your answers, Q1. Consider the system -*-y-20 x+y-z=5 a) Find the determinant of the coefficient matrix A. b) Solve the system using the Cramer's rule. c) Determine if the Cramer's rule can be applied the following system or not? Justify. (No need to solve the system x+y+z=7 x-y3 x-y+

Answer 1

a) Determinant of coefficient matrix will be 1 -1 -1 -1 A= 1 1 1 -1 upon expending along row 1 A= -1(1(-1) – 1*(-1)) - (-1)(-1*(-1) – 1*(-1))+(-1)(-1*1 – 1x1) = -1(-1 + 1) + 1(1 + 1) – 1(-1 – 1) = 0 +2 +2 = 4

u b) Using Cramer's rule: We will find 41, 42 and 43, where 4x, Ay and Az, can be found by replacing column 1, 2 and 3 of A respectively by the constant column. Ax thus, value of x = y = and z= Δ -1 Ax = 10 6 5 1 E1- 1 1 0 Ay -1 6 -1 = 0 - (-1)(6 (-1) – 5 X(-1))+(-1)(61 – 5 x 1) -1 = 0 -1-1= -2 -1 -1 = (-1)(6 *(-1) – 5 *(-1)) – 0+(-1)((-1) * 5 – 1x6) -11 = 1-0 + 11 = 12 -1 0 1 6 = (-1)(5 – 6)-(-1)(-5 – 6) + 0 = 1 – 11 + 0 = -10 51 1 5 -1 = -1 Ax 1 1 So, -2 = = 4 12 Ax X = Δ Ay y = Δ Az Z = = = T~ ~ Ya 10 4 2 is the required solution.

c) Cramer's will not be applicable when determinant of coefficient matrix is equal to zero. Here, determinant of coefficient matrix will be 1 1 s=1 1 1 A= -1 -1 expanding along row 1 = 1((-1) *1-(-1) * 1) – 1(1x1–1x1) +1(1^(-1) – 1*(-1)) = 1 x 0 - 1 0 +1x0 = 0 * So, Cramer's rule is not applicable. *also we can say directly determinant is Zero because column 1 and 3 are identical.

I hope this helped.