Chapter Review 341
27. The matrix M and the vectors V and w are given by
M=12 [4
p.
q
2J
L7J
L5J
9. Given that pT denotes the transpose of the matrix P, which one of the following statements concerning 2 x 2 matrices and their determinants may be false?
A. det2P=4detP
iJ.
a) Given that the matrix N is the inverse of M,
B. det(PQ) = det P det Q
ii) state the value ofp if N is singular. b) Prove thatM(v — w) =M(v) — M(w). c) Given M(v — w) = v + w, calculate the
E. PQ)T = QTPT. (83 H> *see page 362
i) write down N, and
values of p and q. d) Given det (M2) = 64, calculate the values of p. (86 SMS
C. det(P+Q)=detP+detQ D. detPr=detP
30. A linear transformation L is such that 131 101 1—ili.FindLi1 2 Li 1=1 iandLi 1=1 Lii L 2J L—1 LOJ Lii
Iii
.
(85 H)
28.
and w are three vectors in a two-dimensional rectangular Cartesian coordinate system with origin o. i and j are unit vectors in the direction of the coordinate axes. A, B and C are three points such that OA = u = 2 i — OB = v = + 3j
i
OC = w
= 7i +
j
2
a) Show A, B and Con a diagram. b) Find the values of )L and such that w = )u + uv. c) Prove that BA and BC are perpendicular. d) Find the coordinates of D so that ABCD is a rectangle and determine the magnitudes of the vectors BA and BC. e) The rectangle ABCD is transformed to the quadrilateral A'B'C'D' under the
transformation with matrix
1 —1 2
[—1
i) Calculate the coordinates of the points A', B', C' and D'. ii) Prove that A'B'C'D' is a parallelogram. iii) Calculate the value of a, a that BB' = aBC'. (83 SMS)
oi, such
3 I. Oxy is a 2-dimensional rectangular Cartesian coordinate system. Points of the system P(x,y) are mapped onto points P'(x',y') by a linear transformation T represented by the (2 x 2) matrix M, in such a way that the coordinates obey the relation
[;:] = M[x], where M=I1.4 —0.2 L0.8
0.6
a) Find the images under T of the points 0, A, B, C whose coordinates are (0,0), (1,2), (3,1) and (2,—i) respectively.
b) Draw a sketch on squared paper
showing the figure OABC and its image O'A'B'C'.
c) Prove that all points on the line with equation y = 2x are invariant under T. d) Describe fully the geometrical effect of the linear transformation T. e) Determine the images of the points A and C under the linear transformation 7—i.
(88 S)