Question

Write the augmented matrix of the given system of equations. $$ \left\\{\begin{array}{r} x+y-z=2 \\ 3 x-2 y=2 \\ 5 x+3 y-z=1 \end{array}\right. $$

Short answer

The augmented matrix is \(\begin{bmatrix} 1 & 1 & -1 & | & 2 \ 3 & -2 & 0 & | & 2 \ 5 & 3 & -1 & | & 1 \ \end{bmatrix}\.

Step-by-step solution

Identify the Coefficients and Constants

First, identify the coefficients of each variable in the system of equations and the constants on the right-hand side of the equations.

Form the Coefficient Matrix

Construct a matrix where each row contains the coefficients of the respective equation. The order of the columns should be the same as the order of the variables in each equation.

Add the Constants Matrix

Append the constants from the right-hand side of each equation to the coefficient matrix. This forms the augmented matrix.

Write the Augmented Matrix

Combine the coefficient matrix and the constants into a single augmented matrix with a vertical bar separating them.

Key concepts

systems of equations

A system of equations includes multiple equations that share common variables.
For example, the original problem has three equations involving the variables \( x \), \( y \), and \( z \).
In a system of equations, we seek values for these variables that make all the equations true at the same time.
The augmented matrix technique is a powerful tool to solve these systems efficiently.

coefficient matrix

The coefficient matrix consists of only the coefficients of the variables in the system of equations.
For example, in the given system:

• \( x + y - z = 2 \)
• \( 3x - 2y = 2 \)
• \( 5x + 3y - z = 1 \)

the coefficients are 1, 1, -1, 3, -2, 0, 5, 3 and -1.
By placing these coefficients into rows and columns based on their variables, we form the coefficient matrix:
\[ \begin{bmatrix} 1 & 1 & -1 \ 3 & -2 & 0 \ 5 & 3 & -1 \end{bmatrix} \]

linear algebra

Linear algebra is a branch of mathematics concerning linear equations and their representations through matrices and vector spaces.
It provides methods to solve systems of equations, perform operations on matrices, and understand the properties of vector spaces.
In the context of our problem, linear algebra helps us convert the system of equations into an augmented matrix, making it easier to solve using techniques like Gaussian elimination or matrix inversion.

matrix representation

Matrix representation is a way of organizing linear equations into matrix form.
This allows for more straightforward operations and manipulation using matrix algebra.
To form the augmented matrix for our example: after creating the coefficient matrix:
\[ \begin{bmatrix} 1 & 1 & -1 \ 3 & -2 & 0 \ 5 & 3 & -1 \end{bmatrix} \]
we then add the constants from the right-hand side of each equation as a new column:
\[ \begin{bmatrix} 1 & 1 & -1 & | & 2 \ 3 & -2 & 0 & | & 2 \ 5 & 3 & -1 & | & 1 \end{bmatrix} \]
This forms the augmented matrix, which is essential for solving the system using matrix techniques.