Week Two Review

Inverse Matrices

100

Find the inverse of the following Matrix

[[4,9],[5,8]]

[[-8/13,9/13],[5/13,-4/13]]

200

Find the inverse of the following Matrix

[[-35, 3],[8, 11]]

[[-11/409,3/409],[8/409,35/409]]

300

Find the inverse of the following Matrix

[[-6, -9],[-12,-7]]

[[7/66,-3/22],[-2/11,1/11]]

400

Find the inverse of the following Matrix

[[14, 3],[21, 9]] ( mod 26)

[[15, 21],[17, 6]] ( mod 26)

500

Find the inverse of the following Matrix

[[36,19],[3,10]] (mod 43)

[[5, 12],[20, 18]] (mod 43)

Modular Exponentiation

100

Simplify the following

3^23 (mod 7)

5 ( mod 7)

200

Simplify the following

11^33 (mod 14)

1 (mod 14)

300

Simplify the following

15^63 (mod 123)

69 (mod 123)

400

Simplify the following

17^80 (mod 321)

25 (mod 321)

500

Simplify the following

18^288 (mod 323)

1 (mod 323)

Combinatorics/Abstract Permutations

100

What is the total number of different 12-letter arrangements that can be formed using the letters in the word SUPERCILIOUS?

59875200

200

What is the number of ways to order 9 people in a circle?

40320

300

License plates in New Jersey consist of three letters followed by 4 digits. Assuming that any letter or digit can be repeated, how many different license plates are possible?

175,760,000

400

Find

sigma \odot tau \odot tau \odot sigma

tau = (135)(246), sigma = (21453)(6)

(1)(23654)

500

Let tau = (123)(45678) . What is the minimum number of times  tau  must be composed with itself to result in the identity permutation?

15

Euler Phi Function

100

Calculate 

varphi(7)

6

200

Calculate 

varphi(23)

22

300

Calculate 

varphi(59)

58

400

Calculate 

varphi(145)

112

500

Calculate 

varphi(629)

576