610 Chapter 18 Mathematics in Engineering
(d) [A][B] ?
(e) [A]{C} ?
(f ) Show that [I][A] [A][I] [A]
and
Transpose of a Matrix
As you will see in the classes which you will take later, the formulation and solution of engi-
neering problems lend themselves to situations wherein it is desirable to rearrange the rows of
a matrix into the columns of another matrix.
In general, to obtain the transpose of a matrix [B] of sizemn, the first row of the given
matrix becomes the first column of [B]
T
, the second row of [B] becomes the second column
of [B]
T
, and so on, leading to themth row of [B] becoming themth column of [B]
T
,and
resulting in a matrix with the size ofnm. The matrix [B]
T
reads as a transpose of the [B]
matrix.
Sometimes, in order to save space, we write the solution matrices, which are column matri-
ces, as row matrices using the transpose of the solution — another use for the transpose of a
matrix. For example, we represent the solution given by the Umatrix.3 A 43 I 4 £
050
837
9 29
§£
100
010
001
§£
050
837
9 29
§
3 I 43 A 4 £
100
010
001
§£
050
837
9 29
§£
050
837
9 29
§
3 A 45 C 6 £
050
837
9 29
§£
1
2
5
§•
1021 12 152122 102152
1821 12 132122 172152
1921 12 1 22122 192152
¶•
10
33
32
¶
£
35 10 15
60 75 35
31 77 60
§
£
102142 152172 102112102162 152122 1021321021 22 152132 1021 42
182142 132172 172112182162 132122 1721321821 22 132132 1721 42
192142 1 22172 192112192162 1 22122 1921321921 22 1 22132 1921 42§
3 A 43 B 4 £
050
837
9 29
§£
46 2
72 3
13 4
§
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).圀圀圀⸀夀䄀娀䐀䄀一倀刀䔀匀匀⸀䌀伀䴀圀圀圀⸀夀䄀娀䐀䄀一倀刀䔀匀匀⸀䌀伀䴀