Question

Write a system of linear equations that has each of the following augmented matrices. $$ \text { a. }\left[\begin{array}{rrr|r} 1 & -1 & 6 & 0 \\ 0 & 1 & 0 & 3 \\ 2 & -1 & 0 & 1 \end{array}\right] \quad \text { b. }\left[\begin{array}{rrr|r} 2 & -1 & 0 & -1 \\ -3 & 2 & 1 & 0 \\ 0 & 1 & 1 & 3 \end{array}\right] $$

Short answer

a: \( x - y + 6z = 0, y = 3, 2x - y = 1 \); b: \( 2x - y = -1, -3x + 2y + z = 0, y + z = 3 \).

Step-by-step solution

Understand the Augmented Matrix

An augmented matrix represents a system of linear equations. Each row corresponds to an equation, and each column in front of the bar contains the coefficients of a variable on the left-hand side of the equation. The column after the bar represents the constants on the right-hand side of the equations.

Write the System for Matrix (a)

For matrix (a), - The first row \( \begin{bmatrix} 1 & -1 & 6 \end{bmatrix} \) translates to the equation \( 1x - 1y + 6z = 0 \).- The second row \( \begin{bmatrix} 0 & 1 & 0 \end{bmatrix} \) translates to \( 0x + 1y + 0z = 3 \), which simplifies to \( y = 3 \).- The third row \( \begin{bmatrix} 2 & -1 & 0 \end{bmatrix} \) becomes \( 2x - 1y + 0z = 1 \), or \( 2x - y = 1 \). Thus, the system is:\[ \begin{align*}x - y + 6z &= 0, \y &= 3, \2x - y &= 1\end{align*} \]

Write the System for Matrix (b)

For matrix (b), - The first row \( \begin{bmatrix} 2 & -1 & 0 \end{bmatrix} \) translates to the equation \( 2x - y + 0z = -1 \), or \( 2x - y = -1 \).- The second row \( \begin{bmatrix} -3 & 2 & 1 \end{bmatrix} \) becomes \( -3x + 2y + 1z = 0 \).- The third row \( \begin{bmatrix} 0 & 1 & 1 \end{bmatrix} \) becomes \( 0x + 1y + 1z = 3 \), simplifying to \( y + z = 3 \). Thus, the system is:\[ \begin{align*}2x - y &= -1, \-3x + 2y + z &= 0, \y + z &= 3\end{align*} \]

Key concepts

Augmented Matrix

An augmented matrix is a powerful tool used to represent a system of linear equations efficiently. Imagine having a group of linear equations; an augmented matrix conveniently condenses them into a single structure. This matrix not only holds the coefficients of the variables on one side but also the constants from the equations on the other side. Typically, the matrix is split by a vertical bar, with the left side hosting all variable coefficients while the right side contains the constants.
This layout allows us to easily see the equations' structure and convert between the matrix form and equation form seamlessly.

• The left columns before the vertical line denote the coefficients of variables like x, y, and z.
• The rightmost column, after the vertical line, represents constants from each equation.

For example, in the augmented matrix \[\begin{bmatrix}1 & -1 & 6 & | & 0 \2 & -1 & 0 & | & 1\end{bmatrix}\]- The coefficients before the bar illustrate equations' structures.- The numbers after the bar are the solution or result each equation equates to.

System of Equations

A system of equations is essentially a collection of multiple equations that share the same variables. The purpose of these systems is to identify values for the variables that satisfy all equations simultaneously. Each equation in a system can be represented by a row in its corresponding augmented matrix. This relationship makes augmented matrices particularly useful for organizing and solving systems of equations.To solve a system depicted by an augmented matrix:

• Translate each row into an equation by treating it as the coefficients of the variables followed by the bar and constants.
• Apply methods such as substitution, elimination, or matrix operations (like row reduction) to find solutions.

For instance, consider the system arising from the matrix: \[\begin{bmatrix}1 & -1 & 6 & | & 0 \0 & 1 & 0 & | & 3 \2 & -1 & 0 & | & 1\end{bmatrix}\]This system converts into the equations:- \(1x - 1y + 6z = 0\)- \(0x + 1y + 0z = 3\)- \(2x - 1y + 0z = 1\)By solving these, you determine the intersections or common solutions represented by the variable set that satisfies all.

Matrices

Matrices are structured, rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. They are fundamental in various fields of mathematics and science.
Their uses range from solving linear equations, transforming shapes in graphics, to representing data sets in machine learning. For example, in the world of linear algebra, matrices serve as essential tools for organizing and solving systems of equations.
When a matrix accompanies a system of equations, it maps out the coefficients of each variable in an orderly grid. This structure comprises several components:

• Rows, which correspond to individual equations.
• Columns, which correspond to the coefficients of each variable.

For operations like addition, subtraction, or multiplication, matrices follow specific rules that allow them to be manipulated and solved efficiently. In the context of our exercise, understanding how a matrix simplifies a system of equations can greatly enhance clarity and problem-solving efficiency. Each row encapsulates an equation's coefficients, thus aiding in either manually solving the system or using computational tools for larger, more complex systems.