# Math SL Portfolio: Matricies

Mathematics SL Portfolio Assignment 1

Title: Matrix Powers                                                                                                                           Type 1

1. Consider the matrix M =

Calculate Mn for n = 2, 3, 4, 5, 10, 20, 50.

M²= * = =

M³=*==

M⁴=*==

M⁵=*==

M¹⁰=*==

M²⁰=*==

M⁵⁰=*==

• To obtain the matrices above I multiplied (a*b+b*g), (a*f+b*h), (c*e+d*g), (c*f+d*h)as shown in the general formula seen below

Describe in words any pattern you observe.

In the above matrices the pattern I observed is shown in relationship to the exponents and the numbers within the matrix. As the exponent increases consecutively the matrix is in turn multiplied by two.

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

As a result of the pattern expressed, the General Formula for these matrices is Mⁿ=M*2ⁿ⁻¹

…………………………………………………………………………………………………………………………………………………….

2.  Consider the matrices P =  and S =

Calculate Pn and Sn for other values of n and describe any pattern(s) you observe.

=*==  Determinant: 100-36=64

=*= =  Determinant: 1296-784=512

P⁴=*== Determinant: 18496-14400=4096

P⁵=*=  Determinant: 278784-246016=32768

=*== Determinant: 400-256=144

=*== Determinant: 12544-10816=1728

S⁴=*== Determinant: 430336-409600=20736

S⁵=*= =

Determinant: 15241216-14992384=248832

• To obtain the matrices seen above I used the matrix function on the TI 84 calculator. In order to obtain the discriminate in each problem I used the formula ad –bc such that.

Patterns Found:

The patterns portrayed in matrices Pⁿ and Sⁿ mainly correspond to the coefficient shown in the box and ...