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

Example8.8 IfB =
( ) 123
andC=
456
⎛ ⎞
1
⎝2⎠ findBC.
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8.3 Addition, subtraction and multiplication 263
Solution B has order 2 ×3 andC has order 3 ×1 so clearly the productBC exists and will have
order 2 ×1.BC isformed as follows:
( ) ⎛ ⎞
( ) ( )
123
BC = ⎝1
2⎠ 1 ×1+2×2+3×4 17
=
=
456 4×1+5×2+6×4 38
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Note that the order of the product, 2 ×1, can be determined at the start by considering
the orders ofBandC.
Example8.9 FindABwhere
⎛ ⎞
1 2
⎛ ⎞
−1
A = ⎝ 3 4⎠ and B= ⎝ 2⎠
−1 0
−1
Solution A andBhave orders 3 × 2 and 3 × 1, respectively, and consequently the product,AB,
cannot beformed.
Example8.10 Given
⎛ ⎞
1 11
A = ⎝2 10⎠ and
⎛ ⎞
0 31
B= ⎝4−10⎠
3−12 2 21
find, ifpossible,ABandBA, and comment upon the result.
Solution A andBboth have order 3 × 3 and the productsAB andBA can both be formed. Both
will have order 3 ×3.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 11 0 31 6 42
AB = ⎝2 10⎠
⎝4−10⎠ = ⎝4 52⎠
3−12 2 21 0145
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
0 31 1 11 922
BA = ⎝4−10⎠
⎝2 10⎠ = ⎝234⎠
2 21 3−12 934
ClearlyAB andBA are not the same. Matrix multiplication is not usually commutative
and wemustpay particular attention tothisdetailwhen weareworking withmatrices.
IngeneralAB ≠BA and so matrix multiplication isnot commutative.
In the product AB we say that B has been premultiplied by A, or alternatively A has
beenpostmultiplied byB.