Understanding Engineering Mathematics

In the Applications section (➤403) you are led through an application of eigenvalues
and eigenvectors to the simplest vibrating system there is – a single simple pendulum.
Full details are given on the book website, although it involves some material we have
not covered yet. But be assured, this topic is one of the most important in engineering
mathematics and is well worth the effort to master.

Exercise on 13.7

Find the eigenvalues and corresponding eigenvectors for the matrix

A=

[ 401

− 210

− 201

]

Answer

λ= 1 ,

[ 0

1

0

]

; λ= 2 ,

[− 1

2

2

]

; λ= 3 ,

[− 1

1

1

]

Remember that each of the eigenvectors could be multiplied by a different arbitrary
constant.

13.8 Reinforcement

1.Express the following in matrix form

(i) 2x−y=− 1 (ii) 3x−y+ 2 z= 1

x+ 2 y= 02 x+ 2 y+ 3 z= 2

− 3 x−z=− 3

(iii) a+ 2 b+ 3 c=1(iv)4u− 2 v= 1

a−b−c= 33 u+ 3 v= 2

u−v=− 1

2.IfA=

[− 123

402

− 112

]

B=

[ 013

2 − 14

3 − 12

]

(i) Write downa 12 ,a 31 ,a 33 ,b 11 ,b 21 ,b 32

(ii) Evaluate

∑^3

k= 1

a 1 kbk 3 ,

∑^3

k= 1

a 3 kbk 2

(iii) Evaluate (a) 2A− 3 B (b)A^2 (c)AB (d)BA

3.If

[x − 1 y

203

z 32

]

=

[ 3 ax

b 0 c

23 d

]

evaluatea,b,c,d,x,y,z.