Question 3:
3 | (a) | 3−λ 3 0
det(A−λI)= 0 2−λ 2
1 3 4−λ
=(3−λ)[(2−λ)(4−λ)−2×3]−3(0−2×1) oe
= –λ3 + 9λ2 – 20λ + 12 = 0 | M1
M1
A1
[3] | 1.1a
1.1
1.1 | Formation of appropriate
determinant soi.
Attempt to expand determinant.
Allow one slip.
Must be an equation. ISW. | May see eg expansion by 1st col:
(3−λ)[(2−λ)(4−λ)−6]+1(6−0)
Or other formulation eg:
((3−λ)(2−λ)(4−λ)+6+0)
−(0+6(3−λ)+0)
3 | (b) | 1, 2 and 6 substituted into (a) equation to verify | B1
[1] | 1.1 | eg checking trace is insufficient.
3 | (c) | 3a + 3b = a or 2a or 6a
and 2b + 2c = b or 2b or 6b
and a + 3b + 4c = c or 2c or 6c
λ = 1: 2a = –3b, b = –2c
or λ = 2: c = 0, a = –3b
or λ = 6: a = b, c = 2b
 3  −3 1
     
−2 or 1 or 1
     
     
 1   0  2
 3  −3 1
     
−2 and 1 and 1
     
     
 1   0  2 | M1
M1
A1
A1
[4] | 1.1
1.1
1.1
1.1 | Correctly forming 3 equations in 3
unknowns for one of their
eigenvalues. May see explicit
choice of eg c = 1 to form 3
equations in 2 unknowns.
Attempt to solve equations for at
least one of their eigenvalues
leading to two unknowns in terms
of 3rd.
or any non-zero multiple.
or any non-zero multiple. | Or formation of appropriate
i j k
determinant eg −2 4−λ −4 .
0 −1 3−λ
Attempt to expand determinant
(might be in terms of λ) eg
8−7λ+λ2
 
6−2λ . Can be inferred by
 
 2 
2 correct coefficients.
3 | (d) |  3 −3 1
 
−2 1 1
 
 1 0 2
−1
 3 −3 1 −2 −6 4
  1  
−2 1 1 = −5 −5 5 oe
  10 
 1 0 2  1 3 3
1 0 0 n 1 0 0 
 0 2 0  =  0 2n 0 
   
0 0 6  0 0 6n 
 3 −3 11 0 0  −2 −6 4
 −2 1 1  0 2n 0  1  −5 −5 5 
   10 
 1 0 2  0 0 6n   1 3 3
1 0 0 −2 −6 4
 0 2n 0  −5 −5 5  =
  
 0 0 6n  1 3 3
 −2 −6 4 
 
−5×2n −5×2n 5×2n
 
 6n 3×6n 3×6n
 
 3 −3 1 −2 −6 4 
1   
 −2 1 1−5×2n −5×2n 5×2n=
10  
 1 0 2 6n 3×6n 3×6n 
−6+15×2n+6n −18+15×2n+3×6n 12−15×2n+3×6n
 
1
 4−5×2n+6n 12−5×2n+3×6n −8+5×2n+3×6n
10 
 −2+2×6n −6+6n+1 4+6n+1 
  | M1
A1FT
B1
M1
M1
A1
[6] | 3.1a
3.1a
3.1a
3.1a
1.1
1.1 | Forming matrix of their
eigenvectors, E.
BC. Finding inverse of their
matrix of eigenvectors.
Matrix of eigenvalues must be
consistent with matrix of
eigenvectors. Allow 1n.
Forming EΛnE–1. Can be awarded
if Λn incorrect or uncalculated but
eigenvectors must be in same
order as eigenvalues.
Proper attempt to multiply either
the first two or the last two (of 3)
in the correct order (with or
without 1 ).
10
or
 3 −3×2n 6n  −2 −6 4
 
1  
−2 2n 6n  −5 −5 5 =
 
10  
 1 0 2×6n 1 3 3
 
etc.
Condone 6×6n unsimplified. | May be in decimal form:
−0.2 −0.6 0.4
 
−0.5 −0.5 0.5
 
 0.1 0.3 0.3
 3 −3 11 0 0 
 −2 1 1  0 2n 0  =
  
 1 0 2  0 0 6n 
or
 3 −3×2n 6n 
 
−2 2n 6n 
 2×6n
 1 0 
 