082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

8.11 Eigenvalues and eigenvectors 295
Solving Case 1,forexample by Gaussian elimination, leads tox = 0,y = 0 as the only
possible solution.Thus, the only solution isthe trivialsolution.
Case2
x+y=0
2x+2y =0
This systemwas solved inExample 8.35 usingGaussian elimination toyield
x=−λ,
y=λ
where λ is any number and y is a free variable. Thus there are an infinite number of
solutions.Notethatinthissystemthesecondequation,2x+2y = 0,isamultipleofthe
first equation,x+y = 0.The second equation istwice the first equation.
We now return tothe system
ax+by =0
cx+dy =0
Asseen,dependinguponthevaluesofa,b,candd thesystemhaseitheronlythetrivial
solution or an infinite number of non-trivial solutions. For there to be non-trivial solutionsthesecondequationmustbeamultipleofthefirst.Whenthisisthecase,thencis
a multiple ofaandd is the same multiple ofb, thatis
c = αa, d = αb forsomevalueof α
Inthiscase, consider the quantityad −bc:
ad −bc = a(αb) −b(αa)
= αab− αab
= 0
Hence the condition for non-trivial solutions to exist is thatad −bc = 0. Writing the
systeminmatrix formgives
( ) ( a b x 0
=
c d)(
y 0)
or
where
AX=0
A =
( ) ( ( a b x 0
, X = , 0 =
c d y)
0)
We note that ad −bc is the determinant of A, so non-trivial solutions exist when the
determinant ofAis zero;thatis, whenAisasingular matrix.
Insummary: