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

294 Chapter 8 Matrix algebra
Solutions
1 (a) x=7,y=−6
(b) x=3,y=−5,z=2
(c) x=1−µ,y=2µ,z=µ
(d) x=−3,y=1,z=4
(e) Inconsistent
2 x=2,y=1,z=4
3 (a)
(b)
(c)
1
5
( ) 2 −1
−3 4
⎛ ⎞
5 −5 5
⎝−4 13 −16⎠
3 −6 12
⎛ ⎞
−9 6 −3
⎝−37 22 1⎠
11 −2 1
1
15
1
24
8.11 EIGENVALUESANDEIGENVECTORS
We are now in a position to examine the meaning and calculation of eigenvalues and
theircorrespondingeigenvectors.
8.11.1 Solutiontosystemsoflinearhomogeneousequations
Recall that an equation is linear when the variables occur only to the first power. For
example,
2x+3y=1 (1)
isalinear equation but
2x 2 +3y=1 (2)
isanon-linearequation due tothe term2x 2 .
Equation (1) is called inhomogeneous. When the r.h.s. of a linear equation is 0 then
the equation ishomogeneous.Forexample,
2x+3y=0 and 7x−3y=0
are both homogeneous. This section looks at the solution of systems of linear homogeneous
equations.
Consider the simultaneouslinear homogeneousequations
ax+by =0
cx+dy =0
where a, b, c and d are constants. Clearly x = 0, y = 0 is a solution. It is called the
trivial solution. Non-trivial solutions are solutions other thanx = 0, y = 0. We now
study the system to find conditions on a, b, c and d under which non-trivial solutions
exist.
Fordefinitenessweconsidertwo caseswith values ofa,b,candd given.
Case1
3x−5y =0
6x−7y =0