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 )= 26Problem 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 3Problem 12. Evaluate∣
∣
∣
∣5 ∠ 30 ◦ 2 ∠− 60 ◦
3 ∠ 60 ◦ 4 ∠− 90 ◦∣
∣
∣
∣