Higher Engineering Mathematics, Sixth Edition

236 Higher Engineering Mathematics

∣

∣

∣

∣

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 +j 0 )

=(4−j17.32)or17.78∠− 77 ◦

Now try the following exercise

Exercise 94 Further problems on 2 by 2

determinants

1. Calculate the determinant of

(

3 − 1

− 47

)

[17]

2. Calculate the determinant of(
   − 25
   3 − 6

)

[−3]

3. Calculate the determinant of(
   − 1. 37. 4
   2. 5 − 3. 9

)

[−13.43]

4. Evaluate

∣

∣

∣

∣

j 2 −j 3

( 1 +j) j

∣

∣

∣

∣ [−^5 +j3]

5. Evaluate

∣

∣

∣

∣

∣

2 ∠ 40 ◦ 5 ∠− 20 ◦

7 ∠− 32 ◦ 4 ∠− 117 ◦

∣ ∣ ∣ ∣ ∣ [

(− 19. 75 +j 19. 79 )

or 27. 96 ∠ 134. 94 ◦

]

22.5 The inverse or reciprocal of a 2 by 2 matrix

The inverse of matrixAisA−^1 such thatA×A−^1 =I,

the unit matrix.

Let matrixAbe

(

12

34

)

and let the inverse matrix,A−^1

be

(

ab

cd

)

.

Then, sinceA×A−^1 =I,

(

12

34

)

×

(

ab

cd

)

=

(

10

01

)

Multiplying the matrices on the left hand side, gives

(

a+ 2 cb+ 2 d

3 a+ 4 c 3 b+ 4 d

)

=

(

10

01

)

Equating corresponding elements gives:

b+ 2 d= 0 ,i.e.b=− 2 d

and 3a+ 4 c= 0 ,i.e.a=−

4

3

c

Substituting foraandbgives:

⎛

⎜

⎜

⎜

⎝

−

4

3

c+ 2 c − 2 d+ 2 d

3

(

−

4

3

c

)

+ 4 c 3 (− 2 d)+ 4 d

⎞

⎟

⎟

⎟

⎠

=

(

10

01

)

i.e.

⎛

⎝

2

3

c 0

0 − 2 d

⎞

⎠=

(

10

01

)

showing that

2

3

c=1, i.e.c=

3

2

and− 2 d=1, i.e.d=−

1

2

Sinceb=− 2 d,b=1 and sincea=−

4

3

c,a=−2.

Thus the inverse of matrix

(

12

34

)

is

(

ab

cd

)

that is,

⎛

⎝

− 21

3

2

−

1

2

⎞

⎠

There is, however,a quicker method of obtaining the

inverseof a 2 by 2 matrix.

For any matrix

(

pq

rs

)

the inverse may be

obtained by:

(i) interchanging the positions ofpands,

(ii) changing the signs ofqandr,and

(iii) multiplying this new matrix by the reciprocal of

the determinant of

(

pq

rs

)

Thus the inverse of matrix

(

12

34

)

is

1

4 − 6

(

4 − 2

− 31

)

=

⎛

⎝

− 21

3

2

−

1

2

⎞

⎠

as obtained previously.

Problem 13. Determine the inverse of

(

3 − 2

74

)