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

258 Chapter 8 Matrix algebra
An example of an extremely large electrical network is the national grid in Britain.
Theequationsgoverningthisnetworkareexpressedinmatrixformforanalysisbycomputerbecausesolutionsarerequiredatregularintervalsthroughoutthedayandnightin
order to make decisions such as whether or not a power station should be connected to,
or removed from, the grid.
Toobtainthetrajectoryofarobotitisnecessarytoperformmatrixcalculationstofind
thespeedatwhichvariousmotorswithintherobotshouldoperate.Thisisacomplicated
problemasitisnecessarytoensurethatarobotreachesitsrequireddestinationanddoes
not collide with another object during its movement.
8.2 BASICDEFINITIONS
Amatrix isarectangular pattern or arrayof numbers.
For example,
⎛
A = ⎝
⎞
123
456⎠ B =
−124
( ) 1 2 −2
340.5
C = ( 1 −1 1 )
areallmatrices.Notethatweusuallyuseacapitallettertodenoteamatrix,andenclose
the array of numbers in brackets. To describe the size of a matrix we quote its number
of rows and columns in that order so, for example, an r ×s matrix has r rows and s
columns. We say the matrix hasorderr×s.
Anr×smatrix hasrrows andscolumns.
Example8.1 DescribethesizesofthematricesA,BandCatthestartofthissection,andgiveexamples
of matrices of order 3 ×1,3 ×2 and 4 ×2.
Solution Ahas order 3 ×3,Bhas order 2 ×3 andC has order 1 ×3.
⎛ ⎞
⎛ ⎞
−1
1 2
⎝−2⎠ isa3 ×1 matrix ⎝3 4⎠ isa3 ×2 matrix
3
5 6
and ⎛ ⎞
−1 −1
⎜ ⎜⎝ −1 2
⎟
2 −0.5⎠ 1 0
isa4 ×2 matrix
More generally, ifthe matrixAhasmrows andncolumns wecan write
⎛
⎞
a 11
a 12
... a 1n
a 21
a 22
... a 2n
A = ⎜
⎝
.
.
. .. .
⎟
⎠
a m1
a m2
... a mn