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Date May Example question Marks available 2 Reference code EXM.1.AHL.TZ0.29
Level Additional Higher Level Paper Paper 1 Time zone Time zone 0
Command term Show that Question number 29 Adapted from N/A

Question

Let M(abba)(abba) where aa and bb are non-zero real numbers.

Show that M is non-singular.

[2]
a.

 Calculate M2.

[2]
b.

 Show that det(M2) is positive.

[2]
c.

Markscheme

finding det M =a2+b2=a2+b2         A1

a2+b2>0a2+b2>0, therefore M is non-singular or equivalent statement        R1

[2 marks]

a.

M2 = (abba)(abba)=(a2b22ab2aba2b2)(abba)(abba)=(a2b22ab2aba2b2)           M1A1

[2 marks]

b.

EITHER          

det(M2) =(a2b2)(a2b2)+(2ab)(2ab)=(a2b2)(a2b2)+(2ab)(2ab)                      A1

det(M2) =(a2b2)2+(2ab)2=(a2b2)2+(2ab)2      (=(a2+b2)2)(=(a2+b2)2)

since the first term is non-negative and the second is positive          R1

therefore det(M2) > 0          

Note: Do not penalise first term stated as positive.          

OR          

det(M2) = (det M)2              A1

since det M is positive so too is det (M2)       R1

[2 marks]

c.

Examiners report

[N/A]
a.
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b.
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c.

Syllabus sections

Topic 1—Number and algebra » AHL 1.14—Introduction to matrices
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Topic 1—Number and algebra

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