Finite Mathematics

Matrix is?

a rectangular arrangements of numbers, symbols or expression in rows and columns.

Element

individual number or symbol in a matrix

Rows

Horizontal lines of an element

Columns

Vertical lines of an elements

Dimension

indicates how many rows and columns the matrix has.

Matrix Addition

perform by adding the corresponding elements of two matrices that have the same dimension.

Matrix Subtraction

performed by subtracting the corresponding elements of two matrices with the same dimension

Matrix Multiplication

performed by multiplying two matrices to get new matrix

Scalar Multiplication

multiplying element of matrix by a number

Transpost of a Matrix

obtained by interchanging it rows and columns

Commutative Addition

the order of addition does not affect by the result, still the same

Commutative Addition

A+B = B+A

Associative property of Addition

matrix are group and does not affect the results, still the same

Associative property of Addition

(A+B)+ C = A+(B+C)

Distribute Property

distributing over addition by a scalar(number)

Distribute Property

K(A+B)=KA+KB

Zero Matrix Property

adding zero matrix does not affect the results

Zero Matrix Property

A+0=A

Non-Commutative Property

the order of multiplying matters; its affect the result

Non-Commutative Property

A×B≠B×A

Associative property

grouping does not affect multiplication

Associative property

(A×B)×C=A×(B×C)

Distributive property

multiplication distribute over addition

Distributive property

A×(B+C) = A×B+A×C

Identity Matrix Property

multiplying matrix by an identity Matrix

Identity Matrix Property

A×I=I×A=A

Multiplication by Zero Matrix

multiplying by zero results to zero

Multiplication by Zero Matrix

A×0=0

Transpost of a Transpost

taking the transpost twice and returns to the original matrix

Transpost of a Transpost

(AT)T=A

Transpost of a Sum

transpost of a sum equals the sum of transpost

Transpost of a Sum

(A+B)T = AT+BT

Transpost of a Scalar Multiple

transpost of a matrix multiplied by a scalar

Transpost of a Scalar

(KA)T=KAT

Transpost of a product

transpost of a product reverse the order of multiplication

Transpost of a Product

(AB)T=BTAT

Symmetric Matrices

matrix is symmetry if its equal to its transpost

Symmetric Matrices

A=AT

Matrix A

consist coefficient

Matrix X

contains variables

Matrix B

contains constant number

Linear Equations example

x+y+z=1

Matrix row operation

an action applied to the rows of matrix that transform it without changing the solution of the corresponding system of a linear equations

Row Swapping

Ri—Rj

Row Multiplication

multiplying a non-xero scalar(K) to all the elements within the rows

Row Addition

add a scalar multiple of one row to another row

Elementary Matrix

a matrix form by applying exactly one row operation on identity matrix.

Reduce Row Echelon Form (RREF)

reduce the matrix to make a identity Matrix

Augmented Matrix

a matrix is formed by appending constant matrix B to the matrix A

Augmented Matrix

[A|B]

A= [1 6 4 9]

2 by 2 dimension

A= [ 2 4 6 8 ] B= [ 1 2 5 6 ]

A+B = [ 3 6 11 14]