key term - Augmented matrix

Definition

An augmented matrix is a matrix that represents a system of linear equations, including both the coefficients and the constants from the equations. It combines the coefficient matrix and the constant vector into one larger matrix for easier manipulation and solution.

5 Must Know Facts For Your Next Test

1. An augmented matrix is used to simplify solving systems of linear equations using methods like Gaussian elimination.
2. In an augmented matrix, each row corresponds to one equation in the system, while each column corresponds to a variable or constant term.
3. The vertical bar in an augmented matrix separates the coefficients of variables from the constants on the right-hand side of the equations.
4. Gaussian elimination involves performing row operations on an augmented matrix to transform it into row-echelon form or reduced row-echelon form.
5. An augmented matrix can be used to determine if a system has no solutions, exactly one solution, or infinitely many solutions.

Review Questions

Related terms

Gaussian Elimination:

A method for solving systems of linear equations by transforming the system's augmented matrix into row-echelon form through row operations.

Row-Echelon Form: $$\text{A form of a matrix where all zero rows are at the bottom, and each leading coefficient (the first non-zero number from left) of a non-zero row is to the right of the leading coefficient of the previous row.}$$

Inverse Matrix:

$$\text{A square matrix that, when multiplied by its original matrix, results in an identity matrix. It is used to solve systems of linear equations.}$$