# Matrix Powers

Matrix Powers

1. Consider the matrix M =
1. Calculate Mn for  = 2, 3, 4, 5, 10, 20, 50.

M2 =

M3 =

M4 =

M5 =

M10 =

M20 =

M50 =

1. Describe in words any pattern you observe.

Using a TI-83 Plus to calculate each of these power matrices, we are able to find the th matrix. The matrix provided is an identity matrix, which is uniquely defined by the property In M = M thus the value of Mn is equated to two to the power of  as well; the value zero stays constant for other values of . The matrix Mn  can also be calculated using geometric sequence: Un = U1rn-1;as the common ratio between each matrix is a factor of two.

1. Use this pattern to find a general expression for the matrix Mn in terms of .

Mn = I × 2n

1. Consider the matrices P =  and S =

P2 = 2 =  = 2          S2 = 2 =  = 2

1. 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 ...