An augmented matrix is a mathematical tool that is used to represent a system of linear equations in a compact and organized way. It is a rectangular array of numbers that is used to represent the coefficients of the variables in the equations, as well as the constants on the right-hand side of the equations. This article will explore the definition and meaning of augmented matrix, its origin, associations, synonyms, antonyms, and example sentences.

## Definitions

An augmented matrix is a matrix that represents a system of linear equations. It is formed by placing the coefficients of the variables in the equations in the first columns of the matrix, and the constants on the right-hand side of the equations in the last column of the matrix. The augmented matrix is used to perform operations on the equations, such as row reduction, to solve the system of equations.

## Origin

The concept of augmented matrix was first introduced in the 19th century by mathematicians who were studying linear algebra. The term “augmented” refers to the fact that the matrix is an extension of the coefficient matrix, which only contains the coefficients of the variables in the equations.

## Meaning in different dictionaries

According to the Merriam-Webster dictionary, an augmented matrix is “a matrix obtained by adjoining a column of constants to the coefficients of a system of linear equations.” The Oxford English Dictionary defines it as “a matrix formed by adding a column of constants to the matrix of coefficients of a system of linear equations.”

## Associations

The augmented matrix is commonly used in linear algebra to represent a system of linear equations. It is also used in computer science and engineering to solve problems that involve linear equations, such as optimization problems and control systems.

## Synonyms

Some synonyms of augmented matrix include:

- Extended matrix.
- Augmented coefficient matrix.
- Augmented system of equations.
- Augmented linear system.

## Antonyms

There are no direct antonyms of augmented matrix, but it can be contrasted with the coefficient matrix, which only contains the coefficients of the variables in the system of equations.

## The same root words

The word “augment” comes from the Latin word “augmentare,” which means to increase or enlarge. The word “matrix” comes from the Latin word “mater,” which means mother. Together, these words form the term “augmented matrix,” which means an enlarged or extended mother matrix.

## Example Sentences

- To solve the system of linear equations, we need to write them in the form of an augmented matrix.
- The augmented matrix is used to perform row reduction to solve the system of equations.
- The augmented matrix is a useful tool in linear algebra for representing systems of equations in a compact and organized way.