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.
Verify that
A2-αA+βI= 0.
Assuming that A is non-singular, use the result in part (b)(i) to show that
A-1=1β(αI-A).
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]
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]
multiply throughout by giving M1
0 A1
AG
[2 marks]