Matrix power. The internal assessment will focused on observing patterns of matrix powers which will be the main key to find the general expression of matrix powers.
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Introduction
Internal Assessment- Matrix Powers
The internal assessment will focused on observing patterns of matrix powers which will be the main key to find the general expression of matrix powers. 
1)
Explanation: When matrix M is powered by 2 it gives a result of
from GDC, when M is powered by 3 it gives a result of 
from GDC and when M is powered by 4 it gives a result of 
from GDC. The pattern shown is that every time M is powered by a number after its preceding number it is multiply by 2. For instance, 4 shown in matrix
, 8 shown in matrix
, and 16 shown in matrix 
. Thus, these results make a general expression that
. This formula is proven correct because when
Middle
such as 7 when n = 3, 9 when n = 4 and then 31 when n = 5. Thus, these results make a general expression that
. This formula is proven correct because when
from GDC and this formula also works when
from GDC.
Explanation: When matrix S is powered by 3 it gives a result of 
from GDC, when matrix S is powered by 4 it gives a result of
from GDC, and when matrix S is powered by 5 it gives a result of 
from GDC. The pattern shown is that results have a common factor of 
such as 4 shown in matrix
, 8 shown in matrix
, and then16 shown in matrix
. Once the resulting matrix is factor out, the
Conclusion
is equivalent to result matrix
, where as the results of 
is equivalent to result matrix 
and the results of 
is equivalent to result matrix
. The pattern also shows that whenthe matrix formula is, when the matrix formula is, and when the matrix formula is
. This shows that any number representing 
corresponds to
and
. Thus, these results give a general expression that
. For instance, if 
and 
then 
= 
from GDC.4)
Explanation: After a further investigation with further values of k and n, the results shows that the limitation for n is that 
all negative numbers and non-integers value because when a matrix is powered by negative numbers or non-integers value the calculator gives a math error text. In addition there’s no limitation for k.
5)
In conclusion,
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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