Question

Use Gaussian Elimination to put the given matrix into reduced row echelon form. $$\left[\begin{array}{llllll}2 & 2 & 1 & 3 & 1 & 4 \\ 1 & 1 & 1 & 3 & 1 & 4\end{array}\right]$$

Short answer

The matrix in reduced row echelon form is: \( \begin{bmatrix} 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 3 & 1 & 4 \end{bmatrix} \).

Step-by-step solution

Write the Augmented Matrix

The given system is already represented as an augmented matrix:\[ \begin{bmatrix} 2 & 2 & 1 & 3 & 1 & 4 \ 1 & 1 & 1 & 3 & 1 & 4 \end{bmatrix} \]

Ensure Leading Coefficient is 1 in Row 2

Check that the leading coefficient (first non-zero element from left) of row 2 is 1. This row already has a leading coefficient of 1, so no change is needed yet.

Eliminate First Column Below Leading 1

To eliminate the first column element below the leading 1 in row 2, subtract row 2 multiplied by 2 from row 1:\[ R1 = R1 - 2 \times R2 \]Perform the calculation:\[ \begin{bmatrix} 2 & 2 & 1 & 3 & 1 & 4 \end{bmatrix} - 2 \times \begin{bmatrix} 1 & 1 & 1 & 3 & 1 & 4 \end{bmatrix} = \begin{bmatrix} 0 & 0 & -1 & -3 & -1 & -4 \end{bmatrix} \]The matrix becomes:\[ \begin{bmatrix} 0 & 0 & -1 & -3 & -1 & -4 \ 1 & 1 & 1 & 3 & 1 & 4 \end{bmatrix} \]

Swap Rows

Swap row 1 with row 2 to place the row with a leading 1 at the top of the matrix:\[ \begin{bmatrix} 1 & 1 & 1 & 3 & 1 & 4 \ 0 & 0 & -1 & -3 & -1 & -4 \end{bmatrix} \]

Scale Row 2 to Make Leading Coefficient 1

Divide row 2 by -1 to make the leading coefficient 1:\[ R2 = -1 \times R2 \]Perform the transformation:\[ \begin{bmatrix} 0 & 0 & -1 & -3 & -1 & -4 \end{bmatrix} \to \begin{bmatrix} 0 & 0 & 1 & 3 & 1 & 4 \end{bmatrix} \]The matrix becomes:\[ \begin{bmatrix} 1 & 1 & 1 & 3 & 1 & 4 \ 0 & 0 & 1 & 3 & 1 & 4 \end{bmatrix} \]

Eliminate Column 3 Above Leading 1 in Row 2

Subtract row 2 from row 1 to eliminate the third column element above the leading 1:\[ R1 = R1 - R2 \]Perform the calculation:\[ \begin{bmatrix} 1 & 1 & 1 & 3 & 1 & 4 \end{bmatrix} - \begin{bmatrix} 0 & 0 & 1 & 3 & 1 & 4 \end{bmatrix} = \begin{bmatrix} 1 & 1 & 0 & 0 & 0 & 0 \end{bmatrix} \]The matrix becomes:\[ \begin{bmatrix} 1 & 1 & 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 3 & 1 & 4 \end{bmatrix} \]

Simplify Row 1

Row 1 is already in its simplest form with a leading 1, and no further simplifications are required for this row.The final reduced row echelon form is:\[ \begin{bmatrix} 1 & 1 & 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 3 & 1 & 4 \end{bmatrix} \]

Key concepts

Reduced Row Echelon Form

Reduced Row Echelon Form (RREF) is a simplified version of a matrix obtained after applying Gaussian Elimination. It makes solving and understanding systems of linear equations much easier. Let's break it down further:

The key characteristics of RREF are:

• Each leading entry in a row is 1 and is the only nonzero entry in its column.
• The leading 1 in each row is to the right of the leading 1 in the previous row.
• Rows with all zero entries are at the bottom of the matrix.

This form of a matrix highlights the independence of equations, making the solutions straightforward and easy to interpret. By scaling rows and eliminating nonzero elements using row operations, we ensure every leading coefficient is 1, and each column containing a leading coefficient has zeros elsewhere.

Understanding RREF is crucial in linear algebra as it provides a systematic approach to solve systems of linear equations.

Matrix Operations

When working with matrices, various operations are used to simplify and manipulate them. These operations are integral when converting a matrix to its reduced row echelon form. Here are some vital matrix operations:

• Row Swapping: This process involves switching two rows of a matrix to position a row with a leading coefficient of 1 to the top, aiding in simplification.
• Row Scaling: Each element in a row is multiplied by a non-zero scalar to adjust the leading coefficient to 1, simplifying the equations.
• Row Addition: Involves adding or subtracting multiples of rows to eliminate or introduce coefficients as needed, especially to create zeros in target positions.

Such operations use logical steps to methodically simplify matrices, eventually achieving the reduced row echelon form necessary for solution derivation.

Augmented Matrix

An augmented matrix is a fundamental concept in linear algebra used to represent a system of linear equations. It combines the coefficient matrix with the constants from the equation system:

Imagine a system of linear equations:
\[ ax + by = c \ dx + ey = f \] This system can be represented as:
\[ \begin{bmatrix} a & b & | & c \ d & e & | & f \end{bmatrix} \]
The vertical line denotes the boundary between the coefficients and constants.

This representation simplifies operations like Gaussian elimination, making it easier to handle multiple equations simultaneously. With the augmented matrix, you can apply row operations on both sides of the boundary, which represents performing equivalent operations on the equations they represent.

Linear Algebra

Linear Algebra is a branch of mathematics that primarily concerns itself with vectors, matrices, and linear transformations. It's used in a wide variety of fields including computer science, engineering, and data science because it allows for efficient and concise representation of multiple linear equations.

Some core concepts include:

• Vectors and Matrices: Fundamental components representing quantities and systems.
• Linear Transformations: Functions mapping vectors to vectors, maintaining vector operations.
• Systems of Linear Equations: Solvable using methods like Gaussian elimination and converting matrices to RREF.

Linear Algebra provides the tools to handle the computations needed in these areas, offering insights into problems that involve multiple variables and equations, helping to understand the underlying structures and solutions.