The theory of matrices and determinants 235
- D+E
⎡
⎢
⎣
⎛
⎜
⎝
7 − 18
317
47 − 2
⎞
⎟
⎠
⎤
⎥
⎦
- A−B
[(
− 2 − 3
− 31
)]
- A+B−C
[(
9. 3 − 6. 4
− 7. 516. 9
)]
- 5A+ 6 B
[(
45 7
−26 71
)]
- 2D+ 3 E− 4 F
⎡
⎢
⎣
⎛
⎜
⎝
4. 6 − 5. 6 − 7. 6
17. 4 − 16. 228. 6
− 14. 20. 417. 2
⎞
⎟
⎠
⎤
⎥
⎦
- A×H
[(
− 11
43
)]
- A×B
[(
16 0
−27 34
)]
- A×C
[(
− 6. 426. 1
22. 7 − 56. 9
)]
- D×J
⎡
⎢
⎣
⎛
⎜
⎝
135
− 52
− 85
⎞
⎟
⎠
⎤
⎥
⎦
- E×K
⎡
⎢
⎣
⎛
⎜
⎝
56
12 − 3
10
⎞
⎟
⎠
⎤
⎥
⎦
- D×F
⎡
⎢
⎣
⎛
⎜
⎝
55. 43. 410. 1
− 12. 610. 4 − 20. 4
− 16. 925. 037. 9
⎞
⎟
⎠
⎤
⎥
⎦
- Show thatA×⎡C=C×A
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
A×C=
(
− 6. 426. 1
22. 7 − 56. 9
)
C×A=
(
− 33. 5 − 53. 1
23. 1 − 29. 8
)
Hence they are not equal
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
22.3 The unit matrix
Aunit matrix,I, is one in which all elements of the
leading diagonal (\) have a value of 1 and all other ele-
ments have a value of 0. Multiplication of a matrix by
Iis the equivalent of multiplying by 1 in arithmetic.
22.4 The determinant of a 2 by 2 matrix
Thedeterminantof a 2 by 2 matrix,
(
ab
cd
)
is defined
as (ad−bc).
The elements of the determinant of a matrix are
written between vertical lines. Thus, the determinant
of
(
3 − 4
16
)
is written as
∣
∣
∣
∣
3 − 4
16
∣
∣
∣
∣and is equal to
( 3 × 6 )−(− 4 × 1 ),i.e.18−(− 4 )or 22. Hence the
determinant of a matrix can be expressed as a single
numerical value, i.e.
∣
∣∣
∣
3 − 4
16
∣
∣∣
∣=22.
Problem 10. Determine the value of
∣
∣
∣
∣
3 − 2
74
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
3 − 2
74
∣
∣
∣
∣=(^3 ×^4 )−(−^2 ×^7 )
= 12 −(− 14 )= 26
Problem 11. Evaluate
∣
∣
∣
∣
( 1 +j) j 2
−j 3 ( 1 −j 4 )
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
( 1 +j) j 2
−j 3 ( 1 −j 4 )
∣
∣
∣
∣=(^1 +j)(^1 −j^4 )−(j^2 )(−j^3 )
= 1 −j 4 +j−j^24 +j^26
= 1 −j 4 +j−(− 4 )+(− 6 )
since from Chapter 20,j^2 =− 1
= 1 −j 4 +j+ 4 − 6
=− 1 −j 3
Problem 12. Evaluate
∣
∣
∣
∣
5 ∠ 30 ◦ 2 ∠− 60 ◦
3 ∠ 60 ◦ 4 ∠− 90 ◦
∣
∣
∣
∣