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 A =[abcd].
The matrix B is given by B =[3223].
Show that the eigenvalues of A are real if (a−d)2+4bc⩾0.
Deduce that the eigenvalues are real if A is symmetric.
Determine the eigenvalues of B.
Determine the corresponding eigenvectors.
Markscheme
the eigenvalues satisfy
|a−λbcd−λ|=0 M1
(a−λ)(d−λ)−bc=0 A1
λ2−(a+d)λ+ad−bc=0 A1
the condition for real roots is
(a+d)2−4(ad−bc)⩾0 M1
(a−d)2+4bc⩾0 AG
[4 marks]
if the matrix is symmetric, b = c. In this case, M1
(a−d)2+4bc=(a−d)2+4b2⩾0
because each square term is non-negative R1AG
[2 marks]
the characteristic equation is
λ2−6λ+5=0 M1
λ=1,5 A1
[2 marks]
taking λ=1
[2222][xy]=[00] M1
giving eigenvector =[1−1] A1
taking λ=5
[−222−2][xy]=[00] M1
giving eigenvector =[11] A1
[4 marks]