Understanding Engineering Mathematics

9.Write down the following expansions of the determinant

|A|=

∣

∣

∣

∣

∣

3 − 12

012

4 − 12

∣

∣

∣

∣

∣

(i) By first row

(ii) By second row

(iii) By 3rd column

(iv) By last row

and check that they all lead to the same result.

10.Invert the following matrices

(i)

[

23

14

]

(ii)

[ 231

123

312

]

(iii)

[ 12 − 1

− 112

2 − 11

]

(iv)

[ 234

431

124

]

(v)

[ 120

3 − 14

206

]

11.Solve the equations below by matrix inversion where possible:

(i) x+y+z= 4 (ii) 2x+ 3 y− 4 z=− 15

2 x+ 5 y− 2 z= 33 x− 2 y+ 3 z= 15

x+ 7 y− 7 z= 55 x+ 7 y+ 5 z=− 6

(iii) 2x+ 3 y−z= 5

x−y+ 3 z= 8

3 x+ 4 y− 2 z= 5

12.Solve the equation

∣

∣

∣

∣

∣

x+16 2

1 −x − 5 x− 1

x−14 0

∣

∣

∣

∣

∣

= 0

13.Solve the following systems of equations by Cramer’s rule – when possible:

(i) x+y= 1 (ii) 4x− 12 y= 3

2 x−y= 211 x− 2 y= 1

(iii) x+y+z=1(iv)x−y+ 2 z= 1

x+ 2 y+ 3 z= 0 x+ 2 y− 3 z= 0

x−y−z= 02 x+y−z= 2

14.Decide which of the following systems of equations have non-trivial solutions:

(i) 3x− 2 y= 0 (ii) x+ 4 y= 0

x+y= 02 x+ 8 y= 0