Defintions

Question 1

Multiplication of two matrices using the sigma symbol

Answer 1

1- Let A be an 5 × 3-matrix and B an 3 × 3
2- then by the definition of “multiplication of matrices” we have
3- For each i,j ∈ N with 1 ≤ i ≤ n and 1 ≤ j ≤ m we have:
cij = ai1·b1j + ai2·b2j + ai3·b3j =
4- Thus the multiplication of two matrices via P-symbol:

Question 2

matrices A and B are equal

Answer 2

We say that two matrices A and B are equal if
* A and B are of the same size, i.e. A and B are m × n matrices,
* (i,j)-entry of A is equal to the (i,j)-entry of B, i.e.
aij = bij
for all 1 ≤ i ≤ m, and all 1 ≤ j ≤ n.

Question 3

square matrix

Answer 3

A square matrix is a matrix with the same number of rows and columns.
In other words: An m × n matrix is called a square matrix if m = n
In other words: An m × m matrix is called a square matrix

Question 4

Main diagonal

Answer 4

A square matrix is a matrix with the same number of rows and columns.
In other words: An m × m matrix is called a square matrix
The set of entries xii for all 1 ≤ i ≤ m, is called the main diagonal of X, i.e.
{x11, x22, . . . , xii, . . . , xmm} is the main diagonal of X

Question 5

Identity Matrix

Answer 5

A square matrix (i.e. m × m matrix) is called the identity matrix of order m if
the entries on the main diagonal are equal to one, and equal to zero elsewhere,
i.e. m × m matrix X is identity matrix, denoted by Im if
xii = 1 for all 1 ≤ i ≤ m and xij = 0 if i not equal to j

Question 6

B is an inverse matrix of A

Answer 6

et A be an m × m matrix. A matrix B is called the inverse matrix of A if
A · B = Im and B · A = Im
(in this case B is an m × m matrix and we write B = A−1
)

Question 7

e is identity permutation of order n

Answer 7

The permutation e of order n is called the identity permutation of order n, if e(i) = i
for all i ∈ {1, 2, . . . , n}

Question 8

“τ is the inverse permutation of σ

Answer 8

If τ and σ are permutations of the order n, then we say
τ is the inverse permutation of σ if
σ ◦ τ =and τ ◦ σ =
(
1 2 3 · · · n
1 2 3 · · · n
)

Question 9

“P is the permutation matrix corresponding to a permutation τ

Answer 9

Let permutation τ be of the order n, i.e.
τ =
(
1 2 · · · n
τ (1) τ (2) · · · τ (n)
)
then P is the permutation matrix corresponding to τ (and is denoted by Pτ ) if
* P is n × n matrix, and
* the entry of P in i-th row and in τ (i)-th column is equal to 1, and all other entries
are equal to 0, i.e.
pij =
(
1 if j = τ (i)
0 otherwise

Question 10

“η is permutation corresponding to a matrix P”

Answer 10

Let P be n × n matrix such that there is exactly one one entry 1 in each row and each
column and 0s elsewhere. The permutation η corresponding to P is of the order n,
i.e.
τ =
(
1 2 · · · n
τ (1) τ (2) · · · τ (n)
)
with τ (i) = j if the entry of P in the i-th row and j-th column is 1 for all 1 ≤ i ≤ n

Question 11

” G is closed under *

Answer 11

If for all a, b ∈ G,
a * b = (a creation of an element ) is an element in G then we say that G is closed under

Question 12

” * is an associative operation on G

Answer 12

If for all x, y, z ∈ G we have
(x * y) * z = x *( y * z)
then we say that operation * is associatitive