(a) Find the values of z, zER, for which the matrix
x3 x
9 1
has inverse (marks-2 per part)
x=
x=
x=
(b) Consider the vectors - (3,0) and 7- (5,5).
(i.) Find the size of the acute angle between i and ü. Angle-
(ii). If -(k, 3) is orthogonal to , what is the value of ke k [2 marks]
(c) Let J be the linear transformation from R2 R2 which is a reflection in the horizontal axis followed by a scaling by the factor 2.
(i) If the matrix of J is W y 1₁ what are y and z
y= [2 marks]
z= [2 marks] U N || 62 -H 9 has no inverse. [6 marks-2 per part] [2 marks]
(d) Consider the parallelepiped P in R³ whose adjacent sides are (0,3,0), (3, 0, 0) and (-1,1, k), where k € Z. If the volume of P is 180, find the two possible values of k. [4 marks-2 each]
k=
k=
(e) Given that the vectors = (1,-1,1,-1, 1) and =(-1, k, 1, k, 8) are orthogonal, find the magnitude of . Give your answer in surd form. [3 marks]
v=