Date | May Example question | Marks available | 3 | Reference code | EXM.2.AHL.TZ0.22 |
Level | Additional Higher Level | Paper | Paper 2 | Time zone | Time zone 0 |
Command term | Show that | Question number | 22 | Adapted from | N/A |
Question
Let M2 = M where M = (abcd),bc≠0(abcd),bc≠0.
Show that a+d=1.
Find an expression for bc in terms of a.
Hence show that M is a singular matrix.
If all of the elements of M are positive, find the range of possible values for a.
Show that (I − M)2 = I − M where I is the identity matrix.
Markscheme
Attempting to find M2 M1
M2 = (a2+bcab+bdac+cdbc+d2) A1
b(a+d)=b or c(a+d)=c A1
Hence a+d=1 (as b≠0 or c≠0) AG N0
[3 marks]
a2+bc=a M1
⇒bc=a−a2 A1 N1
[2 marks]
METHOD 1
Using det M = ad−bc M1
det M = ad−a(1−a) or det M = a(1−a)−a(1−a)
(or equivalent) A1
=0 using a+d=1 or d=1−a to simplify their expression R1
Hence M is a singular matrix AG N0
METHOD 2
Using bc=a(1−a) and a+d=1 to obtain bc=ad M1A1
det M = ad−bc and ad−bc=0 as bc=ad R1
Hence M is a singular matrix AG N0
[3 marks]
a(1−a)>0 (M1)
0 < a < 1 A1A1 N3
Note: Award A1 for correct endpoints and A1 for correct inequality signs.
[3 marks]
METHOD 1
Attempting to expand (I − M)2 M1
(I − M)2 = I − 2M + M2 A1
= I − 2M + M A1
= I − M AG N0
METHOD 2
Attempting to expand (I − M)2 = (1−a−b−c1−d)2 (or equivalent) M1
(I − M)2 = ((1−a)2+bc−b(1−a)−b(1−d)−c(1−a)−c(1−d)bc+(1−d)2)
(or equivalent) A1
Use of a+d=1 and bc=a−a2 to show desired result. M1
Hence (I − M)2 = (1−a−b−c1−d) AG N0
[3 marks]