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Date November 2020 Marks available 5 Reference code 20N.1.AHL.TZ0.F_4
Level Additional Higher Level Paper Paper 1 Time zone Time zone 0
Command term Verify Question number F_4 Adapted from N/A

Question

The matrix A is given by A=[abcd].

By considering the determinant of a relevant matrix, show that the eigenvalues, λ, of A satisfy the equation

λ2-αλ+β=0,

where α and β are functions of a, b, c, d to be determined.

[4]
a.

Verify that

A2-αA+βI=0.

[5]
b.i.

Assuming that A is non-singular, use the result in part (b)(i) to show that

A-1=1β(αI-A).

[2]
b.ii.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

|a-λbcd-λ|=0        M1

(a-λ)(d-λ)-bc=0        M1A1

λ2-(a+d)λ+ad-bc=0        A1

α=(a+d); β=ad-bc


[4 marks]

a.

A2=[abcd][abcd]=[a2+bcab+bdac+cdbc+d2]         (M1)A1


A2-(a+d)A+(ad-bc)I=


[a2+bcab+bdac+cdbc+d2]-(a+d)[abcd]+(ad-bc)[1001]        M1


=[a2+bc-a(a+d)+ad-bcab+bd-b(a+d)ac+cd-c(a+d)bc+d2-d(a+d)+ad-bc]         A2


=0         AG


Note: Award A1A0 for a single error.


[5 marks]

b.i.

multiply throughout by A1 giving        M1

A-αI+βA-1=0         A1

A-1=1βαI-A         AG


[2 marks]

b.ii.

Examiners report

[N/A]
a.
[N/A]
b.i.
[N/A]
b.ii.

Syllabus sections

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