THE THEORY OF MATRICES AND DETERMINANTS 271F
11.E×K⎡ ⎢ ⎢ ⎢ ⎢ ⎣
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
31
2612 −2
3
−2
50⎞ ⎟ ⎟ ⎟ ⎟ ⎠
⎤ ⎥ ⎥ ⎥ ⎥ ⎦12.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⎤ ⎥ ⎥ ⎥ ⎥ ⎦25.3 The unit matrix
Aunit matrix,I, is one in which all elements
of the leading diagonal () have a value of 1 and
all other elements have a value of 0. Multiplication
of a matrix byIis the equivalent of multiplying by
1 in arithmetic.
25.4 The determinant ofa2by2
matrix
Thedeterminantofa2by2matrix,
(
ab
cd)
isdefined 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
−j3(1−j4)∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣(1+j) j 2
−j3(1−j4)∣
∣
∣
∣=(1+j)(1−j4)−(j2)(−j3)= 1 −j 4 +j−j^24 +j^26
= 1 −j 4 +j−(−4)+(−6)
since from Chapter 23,j^2 =− 1
= 1 −j 4 +j+ 4 − 6
=− 1 −j 3Problem 12. Evaluate∣
∣
∣
∣5 ∠ 30 ◦ 2 ∠− 60 ◦
3 ∠ 60 ◦ 4 ∠− 90 ◦∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣5 ∠ 30 ◦ 2 ∠− 60 ◦
3 ∠ 60 ◦ 4 ∠− 90 ◦∣
∣
∣
∣=(5∠^30◦)(4∠− 90 ◦)−(2∠− 60 ◦)(3∠ 60 ◦)
=(20∠− 60 ◦)−(6∠ 0 ◦)
=(10−j 17 .32)−(6+j0)
=(4−j17.32)or17.78∠− 77 ◦Now try the following exercise.Exercise 109 Further problems on 2 by 2
determinants- Calculate the determinant of
(
3 − 1
− 47)[17]- Calculate the determinant of
⎛
⎜
⎝1
22
3−1
3−3
5⎞⎟
⎠[
−7
90]- Calculate the determinant of(
− 1. 37. 4
2. 5 − 3. 9
)
[−13.43]- Evaluate
∣
∣
∣
∣j 2 −j 3
(1+j) j∣
∣
∣
∣ [−^5 +j3]- Evaluate
∣
∣
∣
∣2 ∠ 40 ◦ 5 ∠− 20 ◦
7 ∠− 32 ◦ 4 ∠− 117 ◦∣
∣
∣
∣
[
(− 19. 75 +j 19 .79)
or 27. 95 ∠ 134. 94 ◦]