Matrix Calculation - Sharp EL-W531X Operation Manual
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Matrix Calculation
In a certain year (year 0), the share of manufacturer A is 10% and the
share of manufacturer B is 90%. Manufacturer A then releases a new
product, and each following year it maintains 90% of the share a
the previous year (year k), and usurps 20% of the share b
manufacturer B.
Find the transition matrix for this process and the shares of
manufacturers A and B after 5 years.
20%
(Royalties)
Answer
The share of each company after one year is expressed as follows using
a
and b
0
0
a
= 0.9a
1
b
= (1-0.9)a
1
Thus, a
1
a
= 0.9a
2
b
= 0.1a
2
The transition matrix is
A =
In the same way, after two years
a
= 0.9a
2
b
= 0.1a
2
Expressing a
a
= 0.9(0.9a
2
= (0.9 x 0.9 + 0.2 x 0.1)a
= 0.83a
b
= 0.1(0.9a
2
= (0.1 x 0.9 + 0.8 x 0.1)a
= 0.17a
In summary,
a
= 0.83a
2
b
= 0.17a
2
2
A
=
Manufacturer A
Share 10%
.
+ 0.2b
0
0
+ (1-0.2)b
0
and b
are
1
+ 0.2b
1
1
+ 0.8b
1
1
0.9 0.2
0.1 0.8
+ 0.2b
1
1
+ 0.8b
1
1
and b
using a
2
2
+ 0.2b
0
+ 0.34b
0
0
+ 0.2b
0
+ 0.66b
0
0
+ 0.34b
0
0
+ 0.66b
0
0
0.83 0.34
: This is equal to matA
0.17 0.66
10%
Manufacturer B
Share 90%
20%
0
and b
gives
0
0
) + 0.2(0.1a
+ 0.8b
0
0
+ (0.9 x 0.2 + 0.2 x 0.8)b
0
) + 0.8(0.1a
+ 0.8b
0
0
+ (0.1 x 0.2 + 0.8 x 0.8)b
0
27
k
of
k
20%
(Royalties)
)
0
0
)
0
0
2
. (Refer to Example 1)
it had
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