Understanding Engineering Mathematics

4.If

[

cosθ sinθ

−sinθ cosθ

]

=









√

3

2

1

2

−

1

2

√

3

2









determineθin the first quadrant, i.e. 0<θ< 90 °.

5.Using the following matrices evaluate every possible sum and product of pairs of the
matrices (repetitions such asA^2 allowed):

A=

[

2 − 1

30

]

B=

[ 3

− 1

4

]

C=[− 20 ]

D=

[ 01 − 1

203

− 1 − 32

]

E=

[ 34

− 12

03

]

F=

[

111

222

]

G=[ 210 ]

6. A=

[ 2 − 1

30

− 11

]

B=

[ 32 − 1

01 2

11 1

]

C=

[ 42

31

0 − 1

]

D=

[

21 4

30 − 1

]

u=

[− 2

1

− 1

]

v=

[ 1

2

3

]

w=

[− 1

3

2

]

Find, where possible,

(i) 3A+B (ii) 4A+ 2 C (iii) 3D− 2 A

(iv) 2AD+ 3 B (v) 3u− 2 v+B (vi) 2u+ 3 v−w

(vii) u− 2 w+Bv

7.A,B,Care the matrices

A=

[

31

− 12

]

B=

[

2 − 10

312

]

C=

[

112

− 223

]

Verify thatA(B+C)=AB+AC.Isittruethat(B+C)A=BA+CA?

8.Evaluate

(i)

∣

∣

∣

∣

23

− 24

∣

∣

∣

∣ (ii)

∣

∣

∣

∣

∣

102

345

567

∣

∣

∣

∣

∣

(iii)

∣

∣

∣

∣

∣

10 6

3415

5621

∣

∣

∣

∣

∣

(iv)

∣

∣

∣

∣

∣

100

235

413

∣

∣

∣

∣

∣

(v)

∣

∣

∣

∣

∣

023

− 204

− 3 − 40

∣

∣

∣

∣

∣

(vi)

∣

∣

∣

∣

cosθ sinθ

−sinθ cosθ

∣

∣

∣

∣