Question 12:
--- 12(a) ---
12(a) | Forms appropriate equation
using Mλ =λv | AO1.1a | M1 | 1 2 1a a
    
2 2 2 b  4 b
    
1 2 1 c  c
   
5a2b2c0
2a2b2c0
a2b5c0
3a3c=0
1
 
eigenvector is 2
 
1
Eliminates one variable | AO1.1a | M1
Deduces a correct eigenvector | AO2.2a | A1
Q | Marking Instructions | AO | Marks | Typical Solution
(b) | Forms the characteristic
equation of M | AO3.1a | M1 | 1 2 1
2 2 2 0
1 2 1
(1)(2)(1)4 
2(222)1(42)0
3 +12+16=0
(4)(244)0
2420
Eigenvalues are 4, –2, –2
1 2 1x x
    
2 2 2 y  2 y
    
 1 2 1  z  z
     
x + 2y – z = 0
1 1
   
0 and 1
   
1 1 
4 0 0 
 
D 0 2 0
 
0 0 2
 1 1 1
 
U 2 0 1
 
 1 1 1 
Obtains the correct
characteristic equation -
unsimplified | AO1.1b | A1
Obtains roots and identifies
them as eigenvalues for ‘their’
characteristic equation | AO1.1b | A1F
Forms an appropriate matrix
equation using the eigenvalue
–2
FT ‘their’ eigenvalue | AO3.1a | M1
Expands and simplifies to
obtain a single equation in x, y
and z
FT ‘their’ matrix equation
provided both M1 marks have
been awarded | AO1.1b | A1F
Correctly deduces two linearly
independent eigenvectors
CAO | AO2.2a | A1
Correctly identifies that the
matrix D must include 4 and
‘their’ other eigenvalue(s) | AO1.2 | B1F
Correctly identifies the
corresponding U matrix from
‘their’ eigenvectors | AO1.1b | A1F
Total | 11
Q | Marking Instructions | AO | Marks | Typical Solution