Question

Convert the given augmented matrix into a system of linear equations. Use the variables \(x_{1}, x_{2},\) etc. $$\left[\begin{array}{ccccc}1 & 0 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 & 5 \\ 0 & 0 & 0 & 1 & 3\end{array}\right]$$

Short answer

The system is: \( x_1 = 2 \), \( x_2 = -1 \), \( x_3 = 5 \), \( x_4 = 3 \).

Step-by-step solution

Understand the Matrix Structure

The given matrix is in augmented form, with the last column representing the constants of equations.

Identify Variables and Coefficients

The matrix corresponds to a system having four variables: \(x_1, x_2, x_3,\) and \(x_4\). Each row represents an equation where entries before the augmented line (last column) are coefficients, and the entry in the last column is a constant.

Translate Rows to Equations

For each row, write an equation where each element in the row before the last one is multiplied by its respective variable, set equal to the constant in the last column.

Write Equation for Row 1

The first row \([1, 0, 0, 0, 2]\) translates to: \(1 \cdot x_1 + 0 \cdot x_2 + 0 \cdot x_3 + 0 \cdot x_4 = 2\), which simplifies to \(x_1 = 2\).

Write Equation for Row 2

The second row \([0, 1, 0, 0, -1]\) translates to: \(0 \cdot x_1 + 1 \cdot x_2 + 0 \cdot x_3 + 0 \cdot x_4 = -1\), which simplifies to \(x_2 = -1\).

Write Equation for Row 3

The third row \([0, 0, 1, 0, 5]\) translates to: \(0 \cdot x_1 + 0 \cdot x_2 + 1 \cdot x_3 + 0 \cdot x_4 = 5\), which simplifies to \(x_3 = 5\).

Write Equation for Row 4

The fourth row \([0, 0, 0, 1, 3]\) translates to: \(0 \cdot x_1 + 0 \cdot x_2 + 0 \cdot x_3 + 1 \cdot x_4 = 3\), which simplifies to \(x_4 = 3\).

Key concepts

Augmented Matrix

An augmented matrix is a powerful tool used in linear algebra to solve systems of linear equations efficiently. It combines the coefficients of the variables in the system and the constants from the equations into a single matrix form. By aligning these numbers into rows and columns, augmented matrices simplify our work.
- The rows of the matrix represent individual equations.- The columns (except the last one) represent the variables' coefficients.- The last column stores the constants separately, indicating what each equation must equal.
In our example, the augmented matrix has the form: \[\begin{array}{ccccc}1 & 0 & 0 & 0 & 2 \0 & 1 & 0 & 0 & -1 \0 & 0 & 1 & 0 & 5 \0 & 0 & 0 & 1 & 3\end{array}\]
This matrix represents equations of the form where the left-hand side involves coefficients and variables, and the right-hand side equals the constants.

Row Reduction

Row reduction is an essential process to simplify matrices, making them easier to solve or interpret. It's like putting the matrix into its simplest form so that extracting the solution becomes straightforward. This process involves a series of steps:
- **Using row operations** such as row swapping, multiplying a row by a non-zero scalar, and adding or subtracting rows from each other. - **Transforming the matrix** into what's known as the 'row-echelon form' or even further to a 'reduced row-echelon form.' In each step, the goal is to get zeros below the leading coefficients in each row, which are the first non-zero numbers from the left in each row. This method greatly aids in simplifying and solving the system, especially when working with more complex equations. In our example, it's already in its simplified state, where each leading entry is one and other elements in its column are zero.

Linear Algebra

Linear algebra is the branch of mathematics focusing on vectors, matrices, and systems of linear equations. It is central to understanding many concepts related to space and dimension.
- **Vectors:** These are ordered collections of numbers that can represent points in a given space or directions. - **Matrices:** Arrays of numbers arranged in rows and columns; they help in organizing and solving systems of equations.
Using linear algebra, we can model real-world problems and solve them efficiently. It provides the tools for:

• Transforming and manipulating equations.
• Understanding and discovering patterns within mathematical structures like networks, data, and geometric interpretations.

Linear algebra is integral because it simplifies complex systems into a more manageable form, ensuring clear solutions that are easy to communicate.

Solving Equations

Solving equations is the ultimate goal when working with systems of linear equations. Using methods like row reduction, we aim to find a clear solution to each variable. The steps to solving equations often involve:- **Identifying variables**: In the augmented matrix, determine which numbers correlate to each variable.- **Writing equations based on rows**: Convert each row of the augmented matrix back into an equation.- **Solving for each variable**: Apply algebraic operations to isolate and solve for each variable clearly. In our current example, using an augmented matrix enabled us to easily refine the equations and quickly write solutions for \(x_1, x_2, x_3,\) and \(x_4\):

• \(x_1 = 2\)
• \(x_2 = -1\)
• \(x_3 = 5\)
• \(x_4 = 3\)

This systematic approach presents a reliable, quick method for solving and verifying solutions for complex systems.