Higher Engineering Mathematics

274 MATRICES AND DETERMINANTS

Problem 16. Determine the value of

∣

∣

∣

∣

∣

j2(1+j)3

(1−j)1j

0 j 45

∣

∣

∣

∣

∣

Using the first column, the value of the determinant is:

(j2)

∣

∣

∣

∣

1 j

j 45

∣

∣

∣

∣−(1−j)

∣

∣

∣

∣

(1+j)3

j 45

∣

∣

∣

∣

+(0)

∣

∣

∣

∣

(1+j)3

1 j

∣

∣

∣

∣

=j2(5−j^2 4)−(1−j)(5+j 5 −j12)+ 0

=j2(9)−(1−j)(5−j7)

=j 18 −[5−j 7 −j 5 +j^2 7]

=j 18 −[− 2 −j12]

=j 18 + 2 +j 12 = 2 +j 30 or30.07∠86.19◦

Now try the following exercise.

Exercise 111 Further problems on 3 by 3

determinants

1. Find the matrix of minors of
   (
   4 − 76
   − 240
   57 − 4

)

[(

− 16 8 − 34

− 14 − 46 63

− 24 12 2

)]

2. Find the matrix of cofactors of
   (
   4 − 76
   − 240
   57 − 4

)

[(

− 16 − 8 − 34

14 − 46 − 63

− 24 − 12 2

)]

3. Calculate the determinant of
   (
   4 − 76
   − 240
   57 − 4

)

[−212]

4. Evaluate

∣

∣

∣

∣

∣

8 − 2 − 10

2 − 3 − 2

63 8

∣

∣

∣

∣

∣

[−328]

5. Calculate the determinant of
   (
   3. 12. 46. 4
   − 1. 63. 8 − 1. 9


6. 
   
   
   
   33. 4 − 4. 8
   
   
   
   

)

[−242.83]

6. Evaluate

∣

∣

∣

∣

∣

j 22 j

(1+j)1− 3

5 −j 40

∣

∣

∣

∣

∣

[− 2 −j]

7. Evaluate

∣

∣

∣

∣

∣

3 ∠ 60 ◦ j 21

0(1+j)2∠ 30 ◦

02 j 5

∣ ∣ ∣ ∣ ∣ [

26. 94 ∠− 139. 52 ◦or

(− 20. 49 −j 17 .49)

]

8. Find the eigenvaluesλthat satisfy the follow-
   ing equations:

(a)

∣

∣

∣

∣

(2−λ)2

−1(5−λ)

∣

∣

∣

∣=^0

(b)

∣

∣

∣

∣

∣

(5−λ)7 − 5

0(4−λ) − 1

28(− 3 −λ)

∣

∣

∣

∣

∣

= 0

(You may need to refer to chapter 1, pages

8–11, for the solution of cubic equations).

[(a)λ=3or4 (b)λ=1or2or3]

25.7 The inverse or reciprocal of a

3 by 3 matrix

Theadjointof a matrixAis obtained by:

(i) forming a matrixBof the cofactors ofA, and

(ii)transposingmatrixBto giveBT, whereBTis

the matrix obtained by writing the rows ofB

as the columns ofBT. ThenadjA=BT.

Theinverse of matrixA,A−^1 is given by

A−^1 =

adjA

|A|

where adjAis the adjoint of matrixAand|A|is the

determinant of matrixA.