{{ Matrix }}

Matrix- Definition

A matrix is an ordered set of numbers listed rectangular form.
Example. Let A denote the matrix

[2 5 7 8]
[5 6 8 9]
[3 9 0 1]
This matrix A has three rows and four columns. We say it is a 3 x 4 matrix.
We denote the element on the second row and fourth column with a2,4.


Square matrix
If a matrix A has n rows and n columns then we say it's a square matrix.
In a square matrix the elements ai,i , with i = 1,2,3,... , are called diagonal elements.




Diagonal matrix
A diagonal matrix is a square matrix with all de non-diagonal elements 0.
The diagonal matrix is completely denoted by the diagonal elements.
Example.

[7 0 0]
[0 5 0]
[0 0 6]

The matrix is denoted by diag(7 , 5 , 6).



Row matrix
A matrix with one row is called a row matrix.




Column matrix
A matrix with one column is called a column matrix .




Matrices of the same kind
Matrix A and B are of the same kind if and only if
A has as many rows as B and A has as many columns as B.




0-matrixWhen all the elements of a matrix A are 0, we call A a 0-matrix.
We write shortly 0 for a 0-matrix.


An identity matrix I
An identity matrix I is a diagonal matrix with all diagonal element = 1.



A scalar matrix S
A scalar matrix S is a diagonal matrix with all diagonal elements alike.
a1,1 = ai,i for (i = 1,2,3,..n) .




The opposite matrix of a matrix
If we change the sign of all the elements of a matrix A, we have the opposite matrix -A.
If A' is the opposite of A then ai,j' = -ai,j, for all i and j.


A symmetric matrix
A square matrix is called symmetric if it is equal to its transpose.
Then ai,j = aj,i , for all i and j.



A skew-symmetric matrix
A square matrix is called skew-symmetric if it is equal to the opposite of its transpose.
Then ai,j = -aj,i , for all i and j.



The sum of matrices of the same kind
Sum of matrices
To add two matrices of the same kind, we simply add the corresponding elements.
Sum properties
Consider the set S of all n x m matrices (n and m fixed) and A and B are in S.
From the properties of real numbers it's immediate that
• A B is in S
• the addition of matrices is associative in S
• A 0 = A = 0 A
• with each A corresponds an opposite matrix -A
• A B = B A





Multiplication of a matrix
To multiply a matrix with a real number, we multiply each element with this number.
Properties
Consider the set S of all n x m matrices (n and m fixed). A and B are in S; r and s are real numbers.
It is not difficult to see that:

r(A B) = rA rB
(r s)A = rA sA
(rs)A = r(sA)
(A B)T = AT BT
(rA)T = r. AT







Multiplication of a row matrix by a column matrix
This multiplication is only possible if the row matrix and the column matrix have the same number of elements. The result is a ordinary number ( 1 x 1 matrix).
To multiply the row by the column, one multiplies corresponding elements, then adds the results.
Example.

[1]
[2 1 3]. [2] = [19]
[5]




Multiplication of two matrices A.B
This product is defined only if A is a (l x m) matrix and B is a (m x n) matrix.
So the number of columns of A has to be equal to the number of rows of B.
The product C = A.B then is a (l x n) matrix.
The element of the ith row and the jth column of the product is found by multiplying the ith row of A by the jth column of B.

ci,j = sumk (ai,k.bk,j)
Example.

[1 2][1 3] = [5 7]
[2 1][2 2] [4 8]

[1 3][1 2] = [7 5]
[2 2][2 1] [6 6]

[1 1][2 2] = [0 0]
[1 1][-2 -2] [0 0]

From these examples we see that the product is not commutative and that there are zero divisors.





Uses of Matrices
Matrices (singular: matrix, plural: matrices) have many uses in real life. One application would be to use matrices to represent a large amount of data in a concise manner so that we can process the data in various ways more conveniently.
For example, the sales of different types of pre-packed food from 3 stalls during a given period of time could be shown in the form of a table here:
Stall A Stall B Stall C
Packs of noodles sold 36 21 43
Packs of rice sold 27 56 35
This table can be represented as a matrix:

This matrix could then be added with another that represents the sales for a different period of time to get the total for the two periods of time, etc.




What does a matrix consist of ?
A matrix consists of a set of numbers arranged in rows and columns enclosed in brackets.



Elements of the matrix.
Each number in the array is called an entry or an element of the matrix. When we need to read out the elements of an array, we read it out row by row.

Each element is defined by its position in the matrix.
In a matrix A, an element in row i and column j is represented by aij
Example:

a11 (read as ‘a one one ’)= 2 (first row, first column)
a12 (read as ‘a one two') = 4 (first row, second column)
a13 = 5, a21 = 7, a22 = 8, a23 = 9





Types of Matrices :
A matrix may be classified by types. It is possible for a matrix to belong to more than one type.
A row matrix is a matrix with only one row.

E is a row matrix of order 1 × 1

B is a row matrix of order 1 × 3

A column matrix is a matrix with only one column.

C is a column matrix of order 1 × 1

D is a column matrix of order 2 × 1
A column matrix of order 2 ×1 is also called a vector matrix.

A zero matrix or a null matrix is a matrix that has all its elements zero.

O is a zero matrix of order 2 × 3

A square matrix is a matrix with an equal number of rows and columns.

T is a square matrix of order 2 × 2

V is a square matrix of order 3 × 3


A diagonal matrix is a square matrix that has all its elements zero except for those in the diagonal from top left to bottom right; which is known as the leading diagonal of the matrix.

B is a diagonal matrix

A unit matrix is a diagonal matrix whose elements in the diagonal are all ones.

P is a unit matrix.








Identity MatricesA square matrix, I is an identity matrix if the product of I and any square matrix A is A.
IA = AI = A


For a 2 × 2 matrix, the identity matrix for multiplication is
When we multiply a matrix with the identity matrix, the original matrix is unchanged.


If the product of two square matrices, P and Q, is the identity matrix then Q is an inverse matrix of P and P is the inverse matrix of Q. (i.e. PQ = QP = I)
The inverse matrix of A is denoted by A -1. (read as “A inverse”)
AA-1 = A-1A = I
Note that the inverse of A-1 is A.

Example:

Given that B is the inverse of A, find the values of x and y.
Solution:
AB =
Since B is an inverse of A, we know that AB = I

1 – 2y = 1
2y = 0
y = 0
2x = 1
x =

Students:

* Ashwaq Abdulla Al-saqer

* Ahlam al - amri

* Bushra Al-Shmrani

*Monefa Al-Otaybe

* Ahlam al - amri