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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=1βαI-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-λd-λ-bc=0        M1A1

λ2-a+dλ+ad-bc=0        A1

α=a+d; β=ad-bc


[4 marks]

a.

A2=abcdabcd=a2+bcab+bdac+cdbc+d2         (M1)A1


A2-a+dA+ad-bcI=


a2+bcab+bdac+cdbc+d2-a+dabcd+ad-bc1001        M1


=a2+bc-aa+d+ad-bcab+bd-ba+dac+cd-ca+dbc+d2-da+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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