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

298 Chapter 8 Matrix algebra
where λ is some unknown constant. Clearly these equations have the trivial solution
x = 0,y = 0.The equations may be written inmatrix formas
( ) ( 2 1 x x
= λ
3 4)(
y y)
or, usingthe usual notation,
AX=λX
We now seek values of λ so that the system has non-trivial solutions. Although it is
tempting to write (A − λ)X = 0 this would be incorrect sinceA−λ is not defined:A
is a matrix and λ is constant. Hence to progress we need to write the r.h.s. ( in a slightly
x
differentway.Tohelpusdothisweusethe2 ×2identitymatrix,I.Now λ maybe
y)
expressed as
( ) 1 0 x
λ
0 1)(
y
( x
since multiplying by the identity matrix leaves it unaltered. So λX may be written
y)
as λIX. Hence wehave
AX = λIX
which can bewritten as
AX−λIX =0
(A−λI)X =0
Notethattheexpression (A − λI)isdefinedsincebothAand λI aresquarematrices
of the same size.
WehaveseeninSection8.11.1thatforAX = 0tohavenon-trivialsolutionsrequires
|A| = 0.Hence for
(A−λI)X=0
to have non-trivial solutions requires
|A−λI|=0
Now
( ) ( ) 2 1 1 0
A−λI = − λ
3 4 0 1
( ) ( ) 2 1 λ 0
= −
3 4 0 λ
( ) 2 − λ 1
=
3 4−λ
So the condition |A − λI| = 0 gives
∣ 2 − λ 1
3 4−λ∣ = 0