matrix

INTRODUCTION:

 Matrices were formulated and developed as the concept during 18^th and 19^th centuries. Matrices are useful because they enable us to consider an array of many numbers as a single object and perform calculations with these symbols in a very compact form.

DEFINITION OF MATRICES

A matrix is a rectangular array of numbers in row and columns enclosed within a square brackets

TYPES OF MATRICES

 ROW MATRIX

A matrix said to be a row matrix if it has only one row. A row matrix is also called as a row vector.

EXAMPLEA=(5 4 6 7)

COLUMN MATRIX

A matrix is said to be a column matrix if it has only one column. It is also called as a column vector

                          EXAMPLE

A=                       and

B=are the column matrix of order 2*4 and 3*1 respectively

SQUARE MATRIX

A matrix in which the number of rows and the number of columns are equal is said to be a square matrix .

                 EXAMPLE

A=

B=               are square matrices of orders 2 and 3 respectively

DIAGNOAL MATRIX

A diagonal matrix in which all the leading diagonal entries are 1 is called the unit matrix.

EXAMPLE

A=

B=                  are diagonal matrices of orders 2 and 3 respectively.

SCALAR MATRIX

A diagonal matrix in which all the elements along the leading diagonal are equal to a non-zero constant is called a scalar matrix.

 EXAMPLE

A=

B=        are the scalar matrix of order 2 and 3 respectively.

UNIT MATRIX

A diagonal matrix in which all the leading diagonal entries are 1 is called the unit matrix.

EXAMPLE

I =   are unit matrices of orders 2 and 3 respectively.

NULL OR ZERO  MATRIX

A matrix is said to be a null matrix or zero-matrix if each of its elements is zero. It is denoted by o.

EXAMPLE

O=                    are the null matrices of order 2*2

TRANPOSE OF A MATRIX

The transpose of a matrix  A  is obtained by interchanging rows and column of the matrix A and it is denoted by A transpose.

              EXAMPLE

A=  

 Transpose of A=

APPLICATION OF MATRIX

Computer have embedded matrix arithmetic in graphic processing algorithms especially to render reflection and refraction.

The field of probabilities and statistics may use matrix representations.

Before computer graphics, the science of optics used matrix mathematics to account for reflection and for refraction.

 Mathematics, scientists and engineers represent groups of equation as matrices.

Pythagorean formula

Pythagorean theorem

Pythagorean theorem
The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

Geometry
Projecting a sphere to a plane.
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In mathematics, the Pythagorean theorem, also known as Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":^[1]

${\displaystyle a^{2}+b^{2}=c^{2},}$

where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides.

Although it is often argued that knowledge of the theorem predates him the theorem is named after the ancient Greek mathematician Pythagoras (c. 570 – c. 495 BC) as it is he who, by tradition, is credited with its first recorded proof. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework. Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases.

The theorem has been given numerous proofs – possibly the most for any mathematical theorem. They are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps and cartoons abound.

https://en.wikipedia.org/wiki/Pythagorean_theorem