The Row-echelon Form Of The Augmented Matrix Of A System Of Equations Is Given. Find The Solution Of a system of linear equations is a fundamental problem in algebra that involves determining the values of variables satisfying all equations simultaneously. When the augmented matrix of the system is transformed into its row-echelon form, it becomes significantly easier to analyze and solve the system, whether it is consistent or inconsistent, unique or infinite in solutions. This article provides a comprehensive guide on understanding the row-echelon form, the process of transforming an augmented matrix, and methods to find solutions.
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Understanding the Augmented Matrix and Its Significance
What Is an Augmented Matrix?
An augmented matrix is a compact way to represent a system of linear equations. It combines the coefficients of the variables and the constants into a single matrix, facilitating operations like row reduction.Example:
Consider the system:
\[
\begin{cases}
2x + 3y - z = 5 \\
4x + y + 2z = 6 \\
-2x + 5y - 3z = -4
\end{cases}
\]
The augmented matrix is:
\[
\left[\begin{array}{ccc|c}
2 & 3 & -1 & 5 \\
4 & 1 & 2 & 6 \\
-2 & 5 & -3 & -4
\end{array}\right]
\]
Importance of Row Operations
Row operations are used to manipulate the augmented matrix into a form that makes the solution obvious or easier to find. These include:- Row swapping (interchanging two rows)
- Row multiplication (multiplying a row by a non-zero scalar)
- Row addition/subtraction (adding or subtracting multiples of rows)
Row-Echelon Form: Definition and Characteristics
What Is Row-Echelon Form?
A matrix is in row-echelon form if it satisfies the following conditions:- All non-zero rows are above any zero rows.
- The leading coefficient (also called the pivot) of a non-zero row is to the right of the leading coefficient of the row above it.
- The entries below each leading coefficient are zeros.
Why Is Row-Echelon Form Useful?
It simplifies the process of back-substitution to find solutions, especially when dealing with larger systems.---
Transforming the Augmented Matrix into Row-Echelon Form
Step-by-Step Procedure
- Identify the leftmost non-zero column.
- Create a leading 1 (pivot) in the first row by dividing the row by the pivot element.
- Eliminate the entries below the pivot by subtracting suitable multiples of the pivot row.
- Move to the next row and repeat the process for the submatrix, ignoring the rows above.
- Continue until the matrix is in row-echelon form.
Example Transformation
Using the earlier example, perform row operations:Original matrix:
\[
\left[\begin{array}{ccc|c}
2 & 3 & -1 & 5 \\
4 & 1 & 2 & 6 \\
-2 & 5 & -3 & -4
\end{array}\right]
\]
- Divide row 1 by 2 to get a leading 1:
R1 \rightarrow R1/2
\]
\[
\left[\begin{array}{ccc|c}
1 & 1.5 & -0.5 & 2.5 \\
4 & 1 & 2 & 6 \\
-2 & 5 & -3 & -4
\end{array}\right]
\]
- Eliminate below pivot:
R2 \rightarrow R2 - 4 \times R_1
\]
\[
R3 \rightarrow R3 + 2 \times R_1
\]
Calculations:
- \( R_2: (4 - 4 \times 1, 1 - 4 \times 1.5, 2 - 4 \times -0.5, 6 - 4 \times 2.5) = (0, -5, 4, -4) \)
- \( R_3: (-2 + 2 \times 1, 5 + 2 \times 1.5, -3 + 2 \times -0.5, -4 + 2 \times 2.5) = (0, 8, -4, 1) \)
Updated matrix:
\[
\left[\begin{array}{ccc|c}
1 & 1.5 & -0.5 & 2.5 \\
0 & -5 & 4 & -4 \\
0 & 8 & -4 & 1
\end{array}\right]
\]
- Continue similarly to achieve row-echelon form.
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Solving the System Using Row-Echelon Form
Back-Substitution Method
Once the matrix is in row-echelon form, the process of back-substitution involves solving for variables starting from the bottom row upward.Steps:
- Identify the last row, which should contain only one variable.
- Solve for that variable.
- Substitute that value into the above rows.
- Repeat until all variables are found.
Handling Different Types of Systems
- Unique Solution: When the system reduces to a form where each variable can be uniquely determined.
- Infinite Solutions: When there are free variables (variables without leading pivots).
- No Solution: When a row reduces to an inconsistent statement like \( 0 = c \) (where \( c \neq 0 \)).
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Example: Finding the Solution from a Given Row-Echelon Form
Suppose the row-echelon form of an augmented matrix is:
\[
\left[\begin{array}{ccc|c}
1 & 2 & 0 & 3 \\
0 & 1 & -1 & 1 \\
0 & 0 & 0 & 0
\end{array}\right]
\]
This corresponds to the equations:
\[
\begin{cases}
x + 2y + 0z = 3 \\
0x + y - z = 1
\end{cases}
\]
and the third row indicates a free variable.
Solution process:
- From the second equation:
y - z = 1 \Rightarrow y = 1 + z
\]
- From the first equation:
x + 2(1 + z) = 3 \Rightarrow x + 2 + 2z = 3 \Rightarrow x = 1 - 2z
\]
- Let \( z = t \), where \( t \) is any real number (free variable).
Solution set:
\[
\boxed{
\begin{cases}
x = 1 - 2t \\
y = 1 + t \\
z = t
\end{cases}
}
\]
where \( t \in \mathbb{R} \).
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Summary of Key Concepts
- Augmented matrix encapsulates the entire system of linear equations.
- Row-echelon form simplifies the matrix for easier solution extraction.
- Row operations are essential tools for transforming matrices.
- Back-substitution is used to find solutions once the matrix is in row-echelon form.
- Systems can be classified as having a unique solution, infinitely many solutions, or no solution based on the form of the row-echelon matrix.
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Practical Applications of Solving Systems in Row-Echelon Form
Understanding and solving systems of equations using row-echelon form has numerous applications, including:- Engineering (circuit analysis)
- Computer graphics (transformations)
- Economics (equilibrium models)
- Data science (linear regression)
- Operations research (optimization problems)
