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

322 Chapter 8 Matrix algebra
( ) 1 −2
11 SupposeA = .
1 4
( −2
(a) FindAv when v = .
2)
( −2
(b) Find 3vwhen v = .Deduce thatAv = 3v.
2)
( −µ
(c) FindAv when v = forany constant µ.
µ)
Deduce thatAv = 3v.
12 Usethe Gauss--Seidel method to find an approximate
solution of
(a) 5x +3y = −34
2x−7y=93
(b) 3x+y+z=6
2x+5y−z=5
x−3y+8z=14
13 Determine which ofthe followingsystemshave
non-trivial solutions.
(a) 2x−y=0
3x−1.5y=0
(b) 6x+5y=0
5x+6y=0
(c) −x−4y=0
2x+8y=0
(d) 7x−3y=0
1.4x −0.6y = 0
(e) −4x+5y=0
3x−4y=0
14 Determine which ofthe followingsystemshave
non-trivial solutions.
(a) 3x−2y+2z=0
x−y+z=0
2x+2y−z=0
(b) x+3y−z=0
4x−y+2z=0
6x+5y=0
(c) x+2y−z=0
x−3z=0
5x+6y−9z=0
15 ThematrixAis defined by
( ) 3 2
A =
−3 −4
(a) Determine the characteristic equation ofA.
(b) Determine the eigenvalues ofA.
(c) Determine the eigenvectors ofA.
(d) Formanew matrixM whose columns are the two
eigenvectors ofA.M iscalled amodal matrix.
(e) Show thatM −1 AM is adiagonal matrix,D,with
the eigenvalues ofAonits leading diagonal.Dis
called thespectral matrixcorresponding to the
modal matrixM.
16 (a) Show that the matrix
( ) 5 2
A =
−2 1
has only one eigenvalue and determine it.
(b) Calculate the eigenvector ofA.
17 ThematrixH isgiven by
⎛ ⎞
4 −11
H = ⎝−2 40⎠
−4 31
(a) Findthe eigenvalues ofH.
(b) Determine the eigenvectors ofH.
(c) Formanew matrixM whose columns are the
threeeigenvectors ofH.M iscalled a modal
matrix.
(d) ShowthatM −1 HM isadiagonal matrix,D, with
the eigenvalues ofH onits leading diagonal.Dis
called thespectral matrixcorresponding to the
modal matrixM.
Solutions
1 (a) −1 (b) −6
( −5
(c) (−1 −6) (d)
1)
(e) (0 2 2)
⎛ ⎞
7 3
(g) ⎜2
2 1
⎝1
2 0 1 ⎟
⎠
2
(f)
( ) 10 5
15 30
2 1
⎛ ⎞
3 A −1 = 1 0 3 0
⎝0 3 −6⎠
6
2 −2 4
⎛ ⎞
7 −30
A 2 = ⎝ 0 46⎠
−2 23