Higher Engineering Mathematics

270 MATRICES AND DETERMINANTS

In algebra, the commutative law of multiplication
states thata×b=b×a. For matrices, this law is
only true in a few special cases, and in generalA×B
isnotequal toB×A.

Problem 9. IfA=

(

23

10

)

and

B=

(

23

01

)

show thatA×B=B×A.

A×B=

(

23

10

)

×

(

23

01

)

=

(

[(2×2)+(3×0)] [(2×3)+(3×1)]

[(1×2)+(0×0)] [(1×3)+(0×1)]

)

=

(

49

23

)

B×A=

(

23

01

)

×

(

23

10

)

=

(

[(2×2)+(3×1)] [(2×3)+(3×0)]

[(0×2)+(1×1)] [(0×3)+(1×0)]

)

=

(

76

10

)

Since

(

49

23

)

=

(

76

10

)

, thenA×B=B×A

Now try the following exercise.

Exercise 108 Further problems on addition,

subtraction and multiplication of matrices

In Problems 1 to 13, the matricesAtoKare:

A=

(

3 − 1

− 47

)

B=

⎛

⎜

⎝

1

2

2

3

−

1

3

−

3

5

⎞

⎟

⎠

C=

(

− 1. 37. 4

2. 5 − 3. 9

)

D=

(

4 − 76

− 240

57 − 4

)

E=

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

36

1

2

5 −

2

3

7

− 10

3

5

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

F=

(

3. 12. 46. 4

− 1. 63. 8 − 1. 9

5. 33. 4 − 4. 8

)

G=

⎛

⎜

⎝

3

4

1

2

5

⎞

⎟

⎠

H=

(

− 2

5

)

J=

(

4

− 11

7

)

K=

(

10

01

10

)

Addition, subtraction and multiplication

In Problems 1 to 12, perform the matrix opera-

tion stated.

1.A+B

⎡

⎢

⎣

⎛

⎜

⎝

3

1

2

−

1

3

− 4

1

3

6

2

5

⎞

⎟

⎠

⎤

⎥

⎦

2.D+E

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

7 − 16

1

2

33

1

3

7

47 − 3

2

5

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

3.A−B

⎡

⎢

⎣

⎛

⎜

⎝

2

1

2

− 1

2

3

− 3

2

3

7

3

5

⎞

⎟

⎠

⎤

⎥

⎦

4.A+B−C

[(

4. 8 − 7. 73 ̇

− 6. 8 ̇ 310. 3

)]

5. 5A+ 6 B

[(

18. 0 − 1. 0

− 22. 031. 4

)]

6. 2D+ 3 E− 4 F[(
   4. 6 − 5. 6 − 12. 1


7. 4 − 9. 228. 6
   − 14. 20. 413. 0

)]

7.A×H

[(

− 11

43

)]

8.A×B

⎡

⎢

⎣

⎛

⎜

⎝

1

5

6

2

3

5

− 4

1

3

− 6

13

15

⎞

⎟

⎠

⎤

⎥

⎦

9.A×C

[(

− 6. 426. 1

22. 7 − 56. 9

)]

10.D×J

[(

135

− 52

− 85

)]