# Difference Between Vector and Matrix

Vector is a quantity that is represented by a letter with an arrow on top to distinguish it from other mathematical quantities. It represents magnitude and direction.

Matrix is denoted by upper case letters in brackets or parentheses. It is a rectangular array of numbers with elements or entries in it. It has a row vector and column vector which forms a matrix.

**Comparison Table Between Vector and Matrix**

Parameters of Comparison | Vector | Matrix |

Definition | Vector is an array of numbers with elements enclosed in open brackets. | Matrix is a rectangular array of elements or entries in a row and column vector enclosed in open brackets. |

Represents | A Vector represents magnitude and direction in its quantity with units. | A Matrix represents the linear transformations and coefficients of the linear equations. |

Index | Vector has its elements in a single index. | A Matrix has its elements or entries in two indices denoted as row x column. |

Denoted | A Vector is denoted in letters with an arrow on top of it to differentiate it from other quantities. | A Matrix is denoted in upper case letters. |

Uses | A Vector is used in simplifying three-dimensional objects in geometry. | A Matrix is used in linear algebra for linear transformations and forming linear equations. |

**What is Vector?**

Vector is defined as a quantity of an object that has both magnitude and direction. It is denoted by a letter with an arrow on it.

Vector is very important in mathematics, and physics in various domains like linear algebra. A vector can be combined with another vector with its head attached to the other vector's tail.

The addition of two or more vectors results in the same magnitude and direction according to the cumulative and associative law which is the same for the subtraction of vectors as well.

Vector can be used to find the direction of the motion of the object, how gravity is implied on an object, used in oscillators, quantum mechanics, fluid mechanics, in the theory of relativity, the motion of an object across a plane, is used in wave propagation, sound propagation helps in determining the force applied in a three-dimensional object.

**What is Matrix?**

A matrix is a rectangular array of numbers or elements or entries arranged in rows and columns. They are denoted by letters written in upper case.

A Matrix in its plural form is known as Matrices. The size of the matrix is indicated as row x columns, which is written as n x m where n denotes rows and m denotes columns in the matrix.

If the elements above or below the principal diagonal of a square matrix are zero is known as a triangular matrix, if the elements below the principal diagonal are zero then it is known as Upper Triangular Matrix, if the elements above the principal diagonal are zero then it is known as Lower Triangular Matrix.

The matrix in which the number of rows is greater than the number of columns is known as Vertical Matrix, if the number of columns is greater than the number of rows then it is called a Horizontal Matrix.

**Main Differences Between Vector and Matrix**

- Vectors have single index in the rectangular array, whereas Matrix has two indices in their formation.
- Vectors do not change in their magnitude and direction in their mathematical operations, whereas a Matrix change in their magnitude in mathematical operations such as multiplicative operation with respect to associative and cumulative laws.

**Conclusion**

Vector and Matrix are used in mathematics, and physics for transformations and equations. Vectors are used in physics to determine the direction and gravity of the objects in Quantum Mechanics and Fluid Mechanics.

Vector is an array of numbers or elements enclosed in an open bracket that represents the magnitude and direction of the object. Matrix is a rectangular array of numbers in the form of rows and columns with entries and elements representing the magnitude and in mathematical operations they form determinants.

**References**

- https://www.sciencedirect.com/science/article/pii/004269899400257M
- https://pubs.acs.org/doi/pdf/10.1021/ie50548a027

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