Integers and matrices (slides)

FUNDAMENTALS

Properties of IntegersTheorem: If 𝑛𝑛 and 𝑚𝑚 are integers and 𝑛𝑛 > 0, we can uniquely write 𝑚𝑚 = 𝑞𝑞 ⋅ 𝑛𝑛 + 𝑟𝑟 for integers 𝑞𝑞 and 𝑟𝑟 with 0 ≤ 𝑟𝑟 < 𝑛𝑛.

𝑟𝑟 = 𝑚𝑚 mod 𝑛𝑛

Example:

𝑛𝑛 = 3, 𝑚𝑚 = 16: 16 = 5 ⋅ 3 + 1

𝑛𝑛 = 10, 𝑚𝑚 = 3: 3 = 0 ⋅ 10 + 3

𝑛𝑛 = 5, 𝑚𝑚 = −11: −11 = (−3) ⋅ 5 + 4

𝑛𝑛 = 5, 𝑚𝑚 = 10: 10 = 2 ⋅ 5 + 02© S. Turaev, CSC 1700 Discrete Mathematics

Prime NumbersDefinition: A positive integer 𝑝𝑝 > 1 is called prime, if the only positive integers that divide 𝑝𝑝 are 𝑝𝑝 and 1.

Example: 2, 3, 5, 7, 11, 13, … are prime.


Is 1 prime?

Prime NumbersAlgorithm for determining if an integer 𝑛𝑛 > 1 is prime:

Step 1: Check if 𝑛𝑛 is 2. if so, 𝑛𝑛 is prime. If not, proceed to

Step 2: Check if 2|𝑛𝑛. if so, 𝑛𝑛 isn’t prime. If not, proceed to

Step 3: Compute the largest integer 𝑘𝑘 ≤ 𝑛𝑛; proceed to

Step 4: Check if 𝑑𝑑|𝑛𝑛 where 𝑑𝑑 is any odd number in (1, 𝑘𝑘]. If 𝑑𝑑|𝑛𝑛, then 𝑛𝑛 isn’t prime; otherwise it is prime.


Prime FactorizationTheorem: Every positive integer 𝑛𝑛 > 1 can be written uniquely as

𝑛𝑛 = 𝑝𝑝1𝑘𝑘1𝑝𝑝2

𝑘𝑘2 ⋯𝑝𝑝𝑠𝑠𝑘𝑘𝑠𝑠

where

𝑝𝑝1 < 𝑝𝑝2 < ⋯ < 𝑝𝑝𝑠𝑠 are distinct prime numbers that divide 𝑛𝑛,

the 𝑘𝑘’s are positive integers giving the number of times each prime occurs as a factor of 𝑛𝑛.

Example: 9 = 3 ⋅ 3 = 32, 36 =, 100 =


Greatest Common DivisorDefinition:

If 𝑎𝑎, 𝑏𝑏 and 𝑘𝑘 are in ℤ+, and 𝑘𝑘|𝑎𝑎 and 𝑘𝑘|𝑏𝑏, we say that 𝑘𝑘 is a common divisor of 𝑎𝑎 and 𝑏𝑏.

If 𝑑𝑑 is the largest such 𝑘𝑘, 𝑑𝑑 is called the greatest common divisor of 𝑎𝑎 and 𝑏𝑏, and we write 𝑑𝑑 =gcd(𝑎𝑎, 𝑏𝑏).

Example:

the common divisors of 12 and 30: 1, 2, 3, 6.gcd 12,30 = 6

gcd 17,95 = 1 (relatively prime numbers)7© S. Turaev, CSC 1700 Discrete Mathematics

Euclidean Algorithm Suppose 𝑎𝑎 > 𝑏𝑏 > 0. We can write

𝑎𝑎 = 𝑘𝑘𝑏𝑏 + 𝑟𝑟

where 𝑘𝑘 ∈ ℤ+ and 0 ≤ 𝑟𝑟 < 𝑏𝑏.

If a number 𝑛𝑛 divides 𝑎𝑎 and 𝑏𝑏, then it must divide 𝑟𝑟, since 𝑟𝑟 = 𝑎𝑎 − 𝑘𝑘𝑏𝑏. Thus,

gcd 𝑎𝑎, 𝑏𝑏 = gcd(𝑏𝑏, 𝑎𝑎 mod 𝑏𝑏)


Euclidean Algorithmgcd 𝑎𝑎, 𝑏𝑏 = gcd(𝑏𝑏, 𝑎𝑎 mod 𝑏𝑏)

divide 𝑎𝑎 by 𝑏𝑏: 𝑎𝑎 = 𝑘𝑘1𝑏𝑏 + 𝑟𝑟1 0 ≤ 𝑟𝑟1 < 𝑏𝑏

divide 𝑏𝑏 by 𝑟𝑟1: 𝑏𝑏 = 𝑘𝑘2𝑟𝑟1 + 𝑟𝑟2 0 ≤ 𝑟𝑟2 < 𝑟𝑟1

divide 𝑟𝑟1 by 𝑟𝑟2: 𝑟𝑟1 = 𝑘𝑘3𝑟𝑟2 + 𝑟𝑟3 0 ≤ 𝑟𝑟3 < 𝑟𝑟2…

divide 𝑟𝑟𝑛𝑛−2 by 𝑟𝑟𝑛𝑛−1: 𝑟𝑟𝑛𝑛−2 = 𝑘𝑘𝑛𝑛𝑟𝑟𝑛𝑛−1 + 𝑟𝑟𝑛𝑛 0 ≤ 𝑟𝑟𝑛𝑛 < 𝑟𝑟𝑛𝑛−1

divide 𝑟𝑟𝑛𝑛−1 by 𝑟𝑟𝑛𝑛: 𝑟𝑟𝑛𝑛−1 = 𝑘𝑘𝑛𝑛+1𝑟𝑟𝑛𝑛 + 𝑟𝑟𝑛𝑛+1 0 ≤ 𝑟𝑟𝑛𝑛+1 < 𝑟𝑟𝑛𝑛

𝑎𝑎 > 𝑏𝑏 > 𝑟𝑟1 > 𝑟𝑟2 > 𝑟𝑟3 > ⋯9© S. Turaev, CSC 1700 Discrete Mathematics

Euclidean Algorithmgcd 𝑎𝑎, 𝑏𝑏 = gcd(𝑏𝑏, 𝑎𝑎 mod 𝑏𝑏)

𝑎𝑎 > 𝑏𝑏 > 𝑟𝑟1 > 𝑟𝑟2 > 𝑟𝑟3 > ⋯ > 𝑟𝑟𝑛𝑛+1 = 0

𝑟𝑟𝑛𝑛−1 = 𝑘𝑘𝑛𝑛+1𝑟𝑟𝑛𝑛 : 𝑟𝑟𝑛𝑛 𝑟𝑟𝑛𝑛−1, 𝑟𝑟𝑛𝑛 𝑟𝑟𝑛𝑛−2, … , 𝑟𝑟𝑛𝑛 𝑟𝑟2, 𝑟𝑟𝑛𝑛 𝑟𝑟1, 𝑟𝑟𝑛𝑛|𝑏𝑏, 𝑟𝑟𝑛𝑛|𝑎𝑎

gcd 𝑎𝑎, 𝑏𝑏 =

gcd 𝑏𝑏, 𝑟𝑟1 = gcd 𝑟𝑟1, 𝑟𝑟2 = ⋯ = gcd 𝑟𝑟𝑛𝑛−1, 𝑟𝑟𝑛𝑛 = 𝑟𝑟𝑛𝑛

Example: Compute gcd(123, 36)


Least Common MultipleDefinition:

If 𝑎𝑎, 𝑏𝑏 and 𝑘𝑘 are in ℤ+, and 𝑎𝑎|𝑘𝑘 and 𝑏𝑏|𝑘𝑘, we say that 𝑘𝑘 is a common multiple of 𝑎𝑎 and 𝑏𝑏.

If 𝑐𝑐 is the smallest such 𝑘𝑘, 𝑐𝑐 is called the least common multiple of 𝑎𝑎 and 𝑏𝑏, and we write 𝑐𝑐 =lcm(𝑎𝑎, 𝑏𝑏).

Example:

the common multiples of 2 and 3: 6, 12, 18, 24, …lcm 2,3 = 6.


Least Common MultipleTheorem: if 𝑎𝑎, 𝑏𝑏 ∈ ℤ+, then

gcd(𝑎𝑎, 𝑏𝑏) ⋅ lcm 𝑎𝑎, 𝑏𝑏 = 𝑎𝑎 ⋅ 𝑏𝑏.

Proof: Use prime factorizations of 𝑎𝑎 and 𝑏𝑏.

Example: Let 𝑎𝑎 = 540 and 𝑏𝑏 = 504. Find gcd and lcm.


Exercises• Write 𝑚𝑚 as 𝑞𝑞𝑛𝑛 + 𝑟𝑟, with 0 ≤ 𝑟𝑟 < 𝑛𝑛:

1. 𝑚𝑚 = 20,𝑛𝑛 = 32. 𝑚𝑚 = 64,𝑛𝑛 = 373. 𝑚𝑚 = 3, 𝑛𝑛 = 12

• Write each integer as a product of powers of primes:828, 1666, 1781

• Find the gcd(𝑎𝑎, 𝑏𝑏) and lcm(𝑎𝑎, 𝑏𝑏):

1. 𝑎𝑎 = 60, 𝑏𝑏 = 1002. 𝑎𝑎 = 77, 𝑏𝑏 = 128


Representation of Integers (Reading)• 3245 means the sum of 3 times 103, 2 times 102, 4

times 10 and 5:

3245 = 3 ⋅ 103 + 2 ⋅ 102 + 4 ⋅ 10 + 5 ⋅ 100

• The base 10 expansion or decimal expansion of 3245

• 10 is called the base of the expansion.

Example:

56893 =


Representation of Integers (Reading)Theorem: if 𝑏𝑏 ∈ ℤ+, then every positive integer 𝑛𝑛 can be uniquely expressed in the form

𝑛𝑛 = 𝑑𝑑𝑘𝑘 ⋅ 𝑏𝑏𝑘𝑘 + 𝑑𝑑𝑘𝑘−1 ⋅ 𝑏𝑏𝑘𝑘−1 + ⋯𝑑𝑑1 ⋅ 𝑏𝑏 + 𝑑𝑑0

where 0 ≤ 𝑑𝑑𝑖𝑖 < 𝑏𝑏, 𝑖𝑖 = 1,2, … , 𝑘𝑘, and 𝑑𝑑𝑘𝑘 ≠ 0.

The sequence 𝑑𝑑𝑘𝑘𝑑𝑑𝑘𝑘−1 ⋯𝑑𝑑1𝑑𝑑0 (more explicitly, 𝑑𝑑𝑘𝑘𝑑𝑑𝑘𝑘−1 ⋯𝑑𝑑1𝑑𝑑0 𝑏𝑏) is called the base 𝑏𝑏 expansion of 𝑛𝑛.

Example: Find the base 2, 3, 4 representations of 173.

Example: Find the decimal expansion of 100111 2.15© S. Turaev, CSC 1700 Discrete Mathematics

MatricesDefinition: A matrix is a rectangular array of numbers in 𝑚𝑚 horizontal rows and 𝑛𝑛 vertical columns:

𝐀𝐀 =

𝑎𝑎11 𝑎𝑎12𝑎𝑎21 𝑎𝑎22 ⋯

𝑎𝑎1𝑛𝑛𝑎𝑎2𝑛𝑛

⋮ ⋮𝑎𝑎𝑚𝑚1 𝑎𝑎𝑚𝑚2 ⋯ 𝑎𝑎𝑚𝑚𝑛𝑛

Definition: The 𝑖𝑖th row of 𝐀𝐀 is [𝑎𝑎𝑖𝑖1 𝑎𝑎𝑖𝑖2 ⋯𝑎𝑎𝑖𝑖𝑛𝑛], 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚,

and the 𝑗𝑗th column of 𝐀𝐀 is

𝑎𝑎1𝑗𝑗𝑎𝑎2𝑗𝑗⋮𝑎𝑎𝑛𝑛𝑗𝑗

, 1 ≤ 𝑗𝑗 ≤ 𝑛𝑛.


MatricesDefinition: We say that 𝐀𝐀 is 𝑚𝑚 by 𝑛𝑛, written 𝑚𝑚 × 𝑛𝑛. If 𝑚𝑚 = 𝑛𝑛, we say 𝐀𝐀 is a square matrix of order 𝑛𝑛. The elements 𝑎𝑎11,𝑎𝑎22, … , 𝑎𝑎𝑛𝑛𝑛𝑛 form main diagonal of 𝐀𝐀.

Definition: We refer to the element 𝑎𝑎𝑖𝑖𝑗𝑗 in the 𝑖𝑖th row and 𝑗𝑗th column of 𝐀𝐀 as the (𝑖𝑖, 𝑗𝑗) entry of 𝐀𝐀. We denote the matrix 𝐀𝐀 by [𝑎𝑎𝑖𝑖𝑗𝑗], for short.

Example:

𝐀𝐀 = 2 3 50 −1 2 ,𝐁𝐁 = 2 6

3 6 ,𝐂𝐂 = 1 0 0 ,𝐃𝐃 =111


MatricesDefinition: A square matrix 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑗𝑗] is called a diagonal matrix if 𝑎𝑎𝑖𝑖𝑗𝑗 = 0 for all 𝑖𝑖 ≠ 𝑗𝑗.

Example:

𝐅𝐅 = 2 00 6 ,𝐆𝐆 =

1 0 00 3 00 0 −3

,𝐇𝐇 =0 0 00 3 00 0 0

Definition: A 𝑚𝑚 × 𝑛𝑛 matrix 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑗𝑗] is called zero matrix, denoted by 𝟎𝟎, if 𝑎𝑎𝑖𝑖𝑗𝑗 = 0 for all 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚, 1 ≤ 𝑗𝑗 ≤ 𝑛𝑛.


MatricesDefinition: Two 𝑚𝑚 × 𝑛𝑛 matrices 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑗𝑗] and 𝐁𝐁 = [𝑏𝑏𝑖𝑖𝑗𝑗]are said to be equal if 𝑎𝑎𝑖𝑖𝑗𝑗 = 𝑏𝑏𝑖𝑖𝑗𝑗 for all 𝑖𝑖 and 𝑗𝑗.

Definition: If 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑗𝑗] and 𝐁𝐁 = [𝑏𝑏𝑖𝑖𝑗𝑗] are 𝑚𝑚 × 𝑛𝑛 matrices, then the sum of 𝐀𝐀 and 𝐁𝐁, denoted by 𝐀𝐀 + 𝐁𝐁, is the matrix 𝐂𝐂 = [𝑐𝑐𝑖𝑖𝑗𝑗] defined by 𝑐𝑐𝑖𝑖𝑗𝑗 = 𝑎𝑎𝑖𝑖𝑗𝑗 + 𝑏𝑏𝑖𝑖𝑗𝑗 for all 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚, 1 ≤ 𝑗𝑗 ≤ 𝑛𝑛.

Example: Find 𝐀𝐀+ 𝐁𝐁 if

𝐀𝐀 =1 0 −83 3 0−2 9 −3

, 𝐁𝐁 =4 5 30 −3 25 0 −2


MatricesTheorem:

𝐀𝐀 + 𝐁𝐁 = 𝐁𝐁 + 𝐀𝐀

𝐀𝐀 + 𝐁𝐁 + 𝐂𝐂 = 𝐀𝐀 + (𝐁𝐁 + 𝐂𝐂)

𝐀𝐀 + 𝟎𝟎 = 𝟎𝟎 + 𝐀𝐀 = 𝐀𝐀


MatricesDefinition: If 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑗𝑗] is 𝑚𝑚 × 𝑝𝑝 matrix and 𝐁𝐁 = [𝑏𝑏𝑖𝑖𝑗𝑗] are 𝑝𝑝 × 𝑛𝑛 matrix, then the product of 𝐀𝐀 and 𝐁𝐁, denotes by 𝐀𝐀𝐁𝐁 is the 𝑚𝑚 × 𝑛𝑛 matrix 𝐂𝐂 = [𝑐𝑐𝑖𝑖𝑗𝑗] defined by

𝑐𝑐𝑖𝑖𝑗𝑗 = 𝑎𝑎𝑖𝑖1𝑏𝑏1𝑗𝑗 + 𝑎𝑎𝑖𝑖2𝑏𝑏2𝑗𝑗 + ⋯𝑎𝑎𝑖𝑖𝑝𝑝𝑏𝑏𝑝𝑝𝑗𝑗

for all 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚, 1 ≤ 𝑗𝑗 ≤ 𝑛𝑛.

Example: Find 𝐀𝐀𝐁𝐁 if

𝐀𝐀 =1 0 −83 3 0−2 9 −3

, 𝐁𝐁 =4 5 30 −3 25 0 −2


MatricesTheorem:

𝐀𝐀(𝐁𝐁𝐂𝐂) = 𝐀𝐀𝐁𝐁 𝐂𝐂

𝐀𝐀 𝐁𝐁 + 𝐂𝐂 = 𝐀𝐀𝐁𝐁 + 𝐀𝐀𝐂𝐂

𝐀𝐀 + 𝐁𝐁 𝐂𝐂 = 𝐀𝐀𝐂𝐂 + 𝐁𝐁𝐂𝐂


MatricesDefinition: A 𝑛𝑛 × 𝑛𝑛 diagonal matrix:

𝐈𝐈𝑛𝑛 =

1 00 1 ⋯ 0

0⋮ ⋮

0 0 ⋯ 1is called the identity matrix of order 𝑛𝑛.

Theorem: For any 𝑛𝑛 × 𝑛𝑛 matrix 𝐀𝐀 and positive integer 𝑝𝑝,

𝐈𝐈𝑛𝑛𝐀𝐀 = 𝐀𝐀𝐈𝐈𝑛𝑛 = 𝐀𝐀

𝐀𝐀𝑝𝑝 = 𝐀𝐀 ⋅ 𝐀𝐀 ⋅ ⋯ ⋅ 𝐀𝐀𝑝𝑝

, 𝐀𝐀0 = 𝐈𝐈𝑛𝑛.


MatricesDefinition: If 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑗𝑗] is 𝑚𝑚 × 𝑛𝑛 matrix, then the 𝑛𝑛 × 𝑚𝑚matrix 𝐀𝐀𝑇𝑇 = [𝑎𝑎𝑖𝑖𝑗𝑗𝑇𝑇 ], where 𝑎𝑎𝑖𝑖𝑗𝑗𝑇𝑇 = 𝑎𝑎𝑗𝑗𝑖𝑖, for all 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚, 1 ≤ 𝑗𝑗 ≤ 𝑛𝑛, is called the transpose of 𝐀𝐀.

Example: Find 𝐀𝐀𝑇𝑇 and 𝐁𝐁𝑇𝑇 if

𝐀𝐀 =1 0 −83 3 0−2 9 −3

, 𝐁𝐁 = 2 −3 56 1 3


MatricesTheorem:

𝐀𝐀𝑇𝑇 𝑇𝑇 = 𝐀𝐀

𝐀𝐀 + 𝐁𝐁 𝑇𝑇 = 𝐀𝐀𝑇𝑇 + 𝐁𝐁𝑇𝑇

𝐀𝐀𝐁𝐁 𝑇𝑇 = 𝐁𝐁𝑇𝑇𝐀𝐀𝑇𝑇

Definition: A matrix 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑗𝑗] is called symmetric if 𝑨𝑨𝑇𝑇=𝐀𝐀.

Definition: if 𝐀𝐀 and 𝐁𝐁 are 𝑛𝑛 × 𝑛𝑛 matrices, we say 𝐁𝐁 is the inverse of 𝐀𝐀 if 𝐀𝐀𝐁𝐁 = 𝐈𝐈𝑛𝑛.


Boolean Matrix OperationsDefinition: A Boolean matrix is 𝑚𝑚 × 𝑛𝑛 matrix whose entries are either zero or one.

Definition: Let 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑗𝑗] and 𝐁𝐁 = [𝑏𝑏𝑖𝑖𝑗𝑗] be 𝑚𝑚 × 𝑛𝑛 Boolean matrices. We define 𝐀𝐀 ∨ 𝐁𝐁 = 𝐂𝐂 = [𝑐𝑐𝑖𝑖𝑗𝑗], the join of 𝐀𝐀 and 𝐁𝐁, by

𝑐𝑐𝑖𝑖𝑗𝑗 = 1 if 𝑎𝑎𝑖𝑖𝑗𝑗 = 1 or 𝑏𝑏𝑖𝑖𝑗𝑗 = 1, 𝑐𝑐𝑖𝑖𝑗𝑗 = 0 if 𝑎𝑎𝑖𝑖𝑗𝑗 = 𝑏𝑏𝑖𝑖𝑗𝑗 = 0,

and the meet of 𝐀𝐀 and 𝐁𝐁, by

𝑐𝑐𝑖𝑖𝑗𝑗 = 1 if 𝑎𝑎𝑖𝑖𝑗𝑗 = 𝑏𝑏𝑖𝑖𝑗𝑗 = 1, 𝑐𝑐𝑖𝑖𝑗𝑗 = 0 if 𝑎𝑎𝑖𝑖𝑗𝑗 = 0 or 𝑏𝑏𝑖𝑖𝑗𝑗 = 0.


Boolean Matrix OperationsExample: Compute 𝐀𝐀 ∨ 𝐁𝐁 and 𝐀𝐀 ∧ 𝐁𝐁 if

𝐀𝐀 =1 0 10 1 11 1 00 0 0

, 𝐁𝐁 =1 1 01 0 10 0 11 1 0


Boolean Matrix OperationsDefinition: Let 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑗𝑗] be an 𝑚𝑚 × 𝑝𝑝 Boolean matrix and 𝐁𝐁 = [𝑏𝑏𝑖𝑖𝑗𝑗] be a 𝑝𝑝 × 𝑛𝑛 Boolean matrix. The Boolean product of 𝐀𝐀 and 𝐁𝐁, denoted by 𝐀𝐀⊙𝐁𝐁, is the 𝑚𝑚 × 𝑛𝑛matrix 𝐂𝐂 = [𝑐𝑐𝑖𝑖𝑗𝑗] defined by

𝑐𝑐𝑖𝑖𝑗𝑗 = 1 if 𝑎𝑎𝑖𝑖𝑗𝑗 = 1 and 𝑏𝑏𝑖𝑖𝑗𝑗 = 1, for some 1 ≤ 𝑘𝑘 ≤ 𝑝𝑝,

𝑐𝑐𝑖𝑖𝑗𝑗 = 0 otherwise.


Boolean Matrix OperationsExample: Compute 𝐀𝐀⊙𝐁𝐁 if

𝐀𝐀 =1 0 10 1 11 1 0

, 𝐁𝐁 =1 1 01 0 10 0 1


Boolean Matrix OperationsTheorem:

𝐀𝐀 ∨ 𝐁𝐁 = 𝐁𝐁 ∨ 𝐀𝐀

𝐀𝐀 ∧ 𝐁𝐁 = 𝐁𝐁 ∧ 𝐀𝐀

𝐀𝐀 ∨ 𝐁𝐁 ∨ 𝐂𝐂 = 𝐀𝐀 ∨ (𝐁𝐁 ∨ 𝐂𝐂)

𝐀𝐀 ∧ 𝐁𝐁 ∧ 𝐂𝐂 = 𝐀𝐀 ∧ (𝐁𝐁 ∧ 𝐂𝐂)

𝐀𝐀 ∧ 𝐁𝐁 ∨ 𝐂𝐂 = 𝐀𝐀 ∧ 𝐁𝐁 ∨ 𝐀𝐀 ∧ 𝐂𝐂

𝐀𝐀 ∨ 𝐁𝐁 ∧ 𝐂𝐂 = 𝐀𝐀 ∨ 𝐁𝐁 ∧ (𝐀𝐀 ∨ 𝐂𝐂)

𝐀𝐀⊙𝐁𝐁 ⊙ 𝐂𝐂 = 𝐀𝐀⊙ (𝐁𝐁⊙ 𝐂𝐂)30© S. Turaev, CSC 1700 Discrete Mathematics

Integers and matrices (slides)

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Transcript of Integers and matrices (slides)