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

Example8.18 IfA =
Solution We haveA T =
( ) 0 5
, findA
−5 0
T and deduce thatAisskew symmetric.
8.5 Some special matrices 273
( ) 0 −5
which isclearly equal to −A. HenceAisskew symmetric.
5 0
EXERCISES8.5
1 IfA=
(a)findA T ,
(b)findB T ,
(c)findAB,
( ) 3 1
andB=
2 6
(d)find (AB) T ,
( ) −1 4
3 8
(e)deduce that (AB) T =B T A T .
⎛ ⎞
x
2 Treating the columnvectorx = ⎝y⎠asa3×1
z
matrix, findIxwhereI is the 3 ×3 identitymatrix.
( ) a b
3 IfA= showthatAA
c d
T is asymmetric
matrix.
⎛ ⎞ ⎛ ⎞
213 1 −70
4 IfA= ⎝ 421⎠andB=
⎝0 25⎠
−132 3 45
findA T ,B T ,ABand (AB) T .
Deduce that (AB) T =B T A T .
5 Determine the type ofmatrixobtained when two
diagonal matrices are multiplied together.
Solutions
( ) 3 2
1 (a)
1 6
( ) 0 20
(c)
16 56
⎛ ⎞
x
2 ⎝y⎠
z
4
( ) −1 3
(b)
4 8
( ) 0 16
(d)
20 56
⎛ ⎞ ⎛ ⎞
24 −1 103
⎝12 3⎠,
⎝−724⎠,
31 2 055
6 IfA=
( ) a b
is skew symmetric, showthat
c d
a =d = 0,that is the diagonal elementsare zero.
( ) 1 13
7 IfA=
15 7
(a) findA T ,
(b) find (A T ) T ,
(c) deduce that (A T ) T is equal toA.
( ) 9 4
8 IfA=
3 2
(a) findA +A T andshowthat thisisasymmetric
matrix,
(b) findA −A T andshowthat thisisaskew
symmetric matrix.
9 The sumofthe elements onthe leading diagonal
ofasquarematrixis known asitstrace. Find the
trace of
( ) ( )
7 2 0 9
(a) (b)
−1 5 −1 0
⎛ ⎞ ⎛ ⎞
(c)
100
⎝010⎠
001
(d)
7 2 1
⎝8 2 3⎠
9 −1 −4
⎛ ⎞ ⎛ ⎞
11 020 11 7 5
⎝ 7 −20 15⎠,
⎝ 0 −20 21⎠
5 2125 20 15 25
5 Diagonal matrix
( ) 1 15
7 (a)
13 7
( ) 18 7
8 (a)
7 4
( ) 1 13
(b)
15 7
( ) 0 1
(b)
−1 0
9 (a)12 (b)0 (c)3 (d)5