(iii) x+y+z=0(iv)x+y+z= 0
x−y+ 2 z= 0 x−y−z= 0
2 x+y− 3 z= 03 x+y+z= 0
(v) 2x− 2 y+z=0(vi)2x−y+z= 0
3 x−y+z= 0 x+y−z= 0
x+y= 03 x+ 2 y+ 2 z= 015.Construct a system of three linear homogeneous equations in three unknowns that has
a non-trivial solution
16.Determine the eigenvalues of the following:
(i)[
10
02]
(ii)[
6 − 3
21]
(iii)[
23
− 1 − 2](iv)[
8 − 4
22]
(v)[− 110
0 − 22
3 − 96]
(vi)[
10
21](vii)[
20 − 1
02 0
−10 2]
(viii)[
12 − 1
21 1
−11 0]
(ix)[
10 0
21 − 2
32 1]17.Determine the eigenvectors for each of the matrices in Q16.
13.9 Applications
1.A standard example in which systems of linear equations in more than one unknown
arise is that of the modelling of electrical circuits ornetworks. A system of elec-
trical components and connections might be very complicated, but can be analysed by
regarding it as comprised of simple loops and branches to whichKirchoff’s Lawscan
be applied. A typical problem might be to determine the current in some part of a
circuit given the resistances, emfs, etc. in the network components. Such a problem
resulted in the following system of equations for currentsi 1 ,i 2 ,i 3 (amps) in a network:
i 1 +i 2 +i 3 = 0− 8 i 2 + 10 i 3 = 0
4 i 1 − 8 i 2 = 6Solve the system fori 1 ,i 2 ,i 3 by (i) Cramer’s rule (ii) matrix inversion.
Notes:- For practical circuits of any size some standard software package would be used,
probably involving numerical methods. - Useful mathematical analysis of the network can be done prior to solution, using the
techniques ofgraph theory, to optimise the solution process - Cramer’s rule and matrix inversion are essentially equivalent.