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

264 Chapter 8 Matrix algebra
Example8.11 Given
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1−12 −1 29 415
A = ⎝ 3 02⎠ B = ⎝ 1 00⎠ C = ⎝231⎠
−1 35 3−21 015
findBC,A(BC),AB and (AB)C, commenting upon the result.
⎛
−1
⎞ ⎛ ⎞
2 9 415
⎛ ⎞
0 14 42
Solution BC = ⎝ 1 00⎠
⎝231⎠ = ⎝4 1 5⎠
3 −2 1 015 8 −2 18
⎛ ⎞ ⎛ ⎞
1 −1 2 0 14 42
⎛
12 9
⎞
73
A(BC) = ⎝ 3 02⎠
⎝4 1 5⎠ = ⎝16 38 162⎠
−1 3 5 8 −218 52 −21 63
⎛ ⎞ ⎛
1 −1 2 −1
⎞
29
⎛
4 −2
⎞
11
AB = ⎝ 3 02⎠
⎝ 1 00⎠ = ⎝ 3 2 29⎠
−1 3 5 3 −21 19 −12 −4
⎛
4 −2
⎞ ⎛ ⎞
11 415
⎛
12 9
⎞
73
(AB)C = ⎝ 3 2 29⎠
⎝231⎠ = ⎝16 38 162⎠
19 −12 −4 015 52 −21 63
We note thatA(BC) = (AB)C so thatinthiscase matrix multiplication isassociative.
The resultobtained inExample 8.11 isalsotrueingeneral:
Matrix multiplication isassociative:
(AB)C =A(BC)
EXERCISES8.3
1 Evaluate
( ) ( )
−1
(a) 1 4
4
(b) ( 3 7 ) ( )
3
−4
( ( ) 5 2 9
(c)
1 3)
1
( ) ( ) 1 4 5 2
(d)
−1 3 −3 4
( ) ( ) 5 1 6 −1
(e)
29 6 −29 5
⎛ ⎞
( )
1
(f) 124 ⎝3⎠
2
⎛ ⎞
(g) ( 5 −1 3 ) 3 3
⎝2 6⎠
4 1
(h) ( 1 9 ) ( )
111
222
( )( ) 1 3 2 1
(i)
−1 0 5 0
( )( ) 3 0 0 3
(j)
0 1 9 0
( ) ( ( )
1 1 2 −7 1
2 IfA= ,B = ,C = ,
3 4 1)
0 4
D = ( 321 ) ( ) 23 4
andE= find,ifpossible,
12 −1
(a)A+D,C−AandD−E
(b) AB,BA,CA,AC,DA,DB,BD,EB,BE andAE
(c) 7C, −3D andkE,wherekisascalar.