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Use this pattern to find a general expression for the matrix Mn in terms of .
Mn = I × 2n
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Consider the matrices P = and S =
P2 = 2 = = 2 S2 = 2 = = 2
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Calculate Pn and Sn for other values of and describe any pattern(s) you observe.
P3 = = 4 S3 = = 4
P4 = = 8 S4 = = 8
P5 = = 16 S5 = = 16
P7 = = 64 S7 = = 64
P10 = = 512
S10 = = 512
From a standard matrix , we can see that the difference between and , and is two. Therefore we could infer that it follows the pattern , and that in matrix P is two whereas in matrix S is three. However, a coefficient had to be factorised in order to have that difference of two. A pattern for the coefficient was found as 2n-1.
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Now consider matrices of the form .
Steps 1 and 2 contain examples of these matrices for = 1, 2 and 3.
Consider other values of , and describe any pattern(s) you observe.
Generalize these results in terms of and .
If =6; M = ; M2 = 2; M3 = 4
If =10; M = ; M2 = 2; M3 = 4
If =-1; M = ; M2 = 2; M3 = 4
If =-2; M = ; M2 = 2; M3 = 4
If =½; M = ; M2 = 2; M3 = 4
There is still a common difference of two between and , and according to 2n-1, ranging from whole values such as six or ten, to negative and rational values. From these results we can generalise the pattern as 2n-1to incorporate other values of .
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Use technology to investigate what happens with further values of and . State the scope or limitations of and .
If =20; M = ; M10 = 512
If =-10; M = ; M-5 = ERR:DOMAIN if put in calculator
M-5 = 0.015265 if put in formula
If =-10; M = ; M0.5 = ERR:DOMAIN if put in calculator
M0.5 = 0.7071067812 if put in calculator
The value of has no limitations, any number plugged into , be it negative, rational, whole, will be coherent with the pattern. The values of however, is limited to only whole numbers. It can go to the extent of whole numbers but it can never be a negative value or a rational value as the error DOMAIN would appear on the calculator. Theoretically, if these disallowed values were to be solely plugged into the pattern 2n-1and not Mn , then similar to the pattern would also function with any rational or negative numbers.
- Explain why your results hold true in general.
Since we have proven the general expression for Mn and all those examples are part of the pattern, the results holds true for all examples in the pattern except for the restrictions that have been proven .
