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

Review exercises 8 321
V = [3;−6;4]
R = [10,−3,−1;−3,14,−2;−1,−2,6]
IPRIME = inv(R) ∗V
Running thisprogram gives
IPRIME = 0.2778
−0.2806
0.6194
REVIEWEXERCISES8
1 Evaluate the following products:
(a) ( −4 1 )( )
2
(b) ( −4 1 )( )
3
7
6
(c) ( −4 1 )( )
( )
2 3 2 −3 −1
(d)
7 6 4 5)(
1
⎛ ⎞
(e) ( 2 −1 0 ) 131 ( )
⎝240⎠
2 1
(f)5
3 6
107
( )
1 732
(g)
2 101
2 Simplify
∣ cosh θ sinh θ
sinh θ cosh θ∣ .
⎛ ⎞
0 23
3 Given thatA = ⎝2 00⎠findA −1 andA 2 .Show
1 −10
thatA 2 +6A −1 −7I = 0, whereI denotes the 3 ×3
identitymatrix.
⎛ ⎞
2 −14
4 IfA= ⎝1 00⎠find
1 −20
(a) |A|
(b) adj(A)
(c) A −1
5 Find the inverse ofthe matrix
⎛ ⎞
1 −20
⎝ 3 15⎠
−1 23
Hencesolvethe equations
x−2y=3
3x+y+5z=12
−x+2y+3z=3
6 Use Gaussian elimination to solve
x+2y−3z+2w=2
2x+5y−8z+6w=5
3x+4y−5z+2w=4
7 Use Jacobi’smethodto obtain a solution ofAX =B
to three decimalplaces where
⎛ ⎞ ⎛ ⎞
10 1 0 1
A = ⎝ 1 10 1 ⎠ and B= ⎝2⎠
0 1 10 1
8 Use amatrixmethodto solve
2x+y−z=3
x−y+2z=1
3x+4y+3z=2
9 Consider the Vandermonde matrix
⎛
1aa 2 ⎞
V = ⎝1bb 2 ⎠
1cc 2
(a) Find detV andshowthat itcanbe written as
(a −c)(a −b)(c −b).
(b) Show thatifa,bandcare all different, thenthe
Vandermonde matrixis non-singular.
10 (a) Thesignal f (t) = sin πt isto be approximated
2
byathird-degree polynomial forvalues oft
between −1 and 2.Byforcingthe original signal
and itsapproximating polynomial to agree at
t=−1,t=0,t=1andt=2,findthis
approximation. [Hint: seeEngineering
application 8.2.]
(b) Useagraphics calculator orgraph plotting
package to compare the graphs of f (t)andits
approximating polynomial.