Practise analysing linear systems, characteristic equations, eigenvectors and diagonal forms, linking matrix calculations to plane geometry, powers and inverses.
It is given that a is a positive constant.
Show that the system of equations
has a unique solution and interpret this situation geometrically.
The matrix A is given by
x=211+a8a2a+5a+10−403−1=4a=0 or y=−21z=−29
M1 A1
Shows that determinant is non-zero or
solves equations and finds at least two
correct values.
The three planes intersect at a single point.
B1
3
Show that the eigenvalues of A are a,-1 and -4.
a−λ2a+5a+10−4−λ03−1−λ=0
M1
Equates determinant to zero.
(a−λ)(−4−λ)(−1−λ)=0⇒λ=a,−1,−4
A1
AG
2
PUBLISHED
Find a matrix P such that
λ=a:ijk0−4−a03−1−a=((−4−a)(−1−a)00)∼(100)
M1 A1
Uses vector product (or equations) to find
corresponding eigenvectors.
λ=−1:ijka+12a+5a+10−30=(3(a+1)0−3(a+1))∼(10−1)
A1
λ=−4:ijka+42a+5a+1033=(3a+12−(3a+12)3a+12)∼(1−11)
A1
P=(1100−10−11)
A1
OE
5
PUBLISHED
Use the characteristic equation of A to find A−1.
(λ−a)(λ+1)(λ+4)=λ3+(5−a)λ2+(4−5a)λ−4a=0
B1
Finds characteristic equation.
4aA−1=A2+(5−a)A+(4−5a)I
M1 A1
Multiplies through by A−1.
A2=(a22a2−17a2−10160−151)
B1
4aA−1=(a22a2−17a2−10160−151)+(5−a)(a2a+5a+10−403−1)+(4−5a)(1001001)
M1
Substitutes for and A in correct equation.
A−1=4a1(45a+84a+40−a0−3a−4a)
A1
6