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Date May 2019 Marks available 4 Reference code 19M.1.AHL.TZ0.F_3
Level Additional Higher Level Paper Paper 1 Time zone Time zone 0
Command term Show that Question number F_3 Adapted from N/A

Question

The matrix A is given by =[abcd].

The matrix B is given by B =[3223].

Show that the eigenvalues of A are real if (ad)2+4bc0.

[4]
a.i.

Deduce that the eigenvalues are real if A is symmetric.

[2]
a.ii.

Determine the eigenvalues of B.

[2]
b.i.

Determine the corresponding eigenvectors.

[4]
b.ii.

Markscheme

the eigenvalues satisfy

|aλbcdλ|=0     M1

(aλ)(dλ)bc=0      A1

λ2(a+d)λ+adbc=0      A1

the condition for real roots is 

(a+d)24(adbc)0      M1

(ad)2+4bc0      AG

[4 marks]

a.i.

if the matrix is symmetric, b = c. In this case,       M1

(ad)2+4bc=(ad)2+4b20

because each square term is non-negative      R1AG

[2 marks]

a.ii.

the characteristic equation is

λ26λ+5=0     M1

λ=1,5      A1

[2 marks]

b.i.

taking λ=1

[2222][xy]=[00]     M1

giving eigenvector =[11]       A1

 

taking λ=5

[2222][xy]=[00]     M1

giving eigenvector =[11]       A1

[4 marks]

b.ii.

Examiners report

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

Syllabus sections

Topic 1—Number and algebra » AHL 1.15—Eigenvalues and eigenvectors
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Topic 1—Number and algebra

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