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

8.3 Addition, subtraction and multiplication 265
3 Plotthe pointsA,B,C with position vectors given by
( ( ( 1 2 2
v 1 = v
0)
2 = v
0)
3 =
3)
respectively. Treating these vectors asmatrices of
order 2 ×1 findthe productsMv 1 ,Mv 2 ,Mv 3 when
( ) 1 0
(a) M=
0 −1
( ) 0 1
(b)M=
1 0
( ) 0 −1
(c) M=
1 0
Ineach casedraw adiagram to illustrate the effect
uponthe vectors ofmultiplication bythe matrix.
4 FindABandBA where
⎛ ⎞
13 2
A = ⎝−10 4⎠
51 −1
⎛ ⎞
521
B = ⎝034⎠
135
5 Given thatA 2 meansthe product ofamatrixAwith
( )
itself,findA 2 4 2
whenA = .FindA
1 3
3 .
( ) ( ) 1 3 2 1
6 IfA= ,B = findAB,BA,A +B
−2 4 −4 5
and (A +B) 2 .Show that
(A+B) 2 =A 2 +AB+BA+B 2
Whyis (A +B) 2 notequaltoA 2 +2AB +B 2 ?
7 Find, ifpossible,
⎛ ⎞⎛
⎞
10 00 1
(a) ⎜00 −10
⎟⎜1
⎟
⎝01 00⎠⎝2⎠
00 01 1
⎛ ⎞⎛
⎞
0010 2
(b) ⎜ 0100
⎟⎜5
⎟
⎝−1000⎠⎝5⎠
0001 1
( ) ⎛ 2 1 36
8 Find ⎝1
3 −5⎠.
2 −57
6 7
⎛ ⎞
1
9 Given the vectorv = ⎝2⎠calculate the vectors
3
obtained whenvispremultiplied by the following
matrices:
⎛ ⎞ ⎛ ⎞
62 9
(a) ⎝ 13 2⎠
−12 −3
⎛ ⎞
131
(c) ⎝926⎠
280
⎛ ⎞
683
(e) ⎜964
⎟
⎝539⎠
252
⎞
(b)
(d)
−103
⎝ 719⎠
134
( ) 312
654
Solutions
1 (a) 15 (b) −19
( ) ( )
47
−7 18
(c) (d)
12
−14 10
( ) 1 0
(e) (f) 15
0 1
(g) (25 12) (h) (19 19 19)
(i)
( ) 17 1
−2 −1
(j)
( ) 0 9
9 0
( ) −8 0
2 (a) A +Ddoes notexist, ,
−3 0
D −E does not exist
( ) 3
(b) ,BAdoes notexist,
10
(c)
3 (a)
(c)
( ) −4 −3
,
12 16
( ) −7 5
,
−21 19
DAdoes notexist,DBdoesnot exist,
( ) 642
,
321
EBdoes notexist,BE doesnot exist,
( ) 3 53
10178
( ) ( )
−49 7 2k 3k 4k
, (−9−6−3),
0 28 k2k −k
( ( ( ) ( ( ( 1 2 2 0 0 3
, , (b) , ,
0)
0)
−3 1)
2)
2)
( ( ( 0 0 −3
, ,
1)
2)
2)