Question

Use Gaussian Elimination to put the given matrix into reduced row echelon form. $$\left[\begin{array}{cccc}1 & 1 & 1 & 2 \\ 2 & -1 & -1 & 1 \\ -1 & 1 & 1 & 0\end{array}\right]$$

Short answer

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

Step-by-step solution

Identify the Leading Entry

The leading entry in the first row is the first non-zero number from the left. In this case, it is '1' in the first row, first column.

Make the Pivot Column Zero Except for Leading Entry

Use the leading '1' in row 1 to eliminate the other non-zero entries below it by applying the operations:1. Replace Row 2 with Row 2 minus 2 times Row 1.2. Replace Row 3 with Row 3 plus Row 1.The matrix becomes:\[\begin{array}{cccc}1 & 1 & 1 & 2 \0 & -3 & -3 & -3 \0 & 2 & 2 & 2\end{array}\]

Normalize the Second Row Pivot

Identify the leading entry in the second row, which is '-3'. Normalize it to '1' by dividing the entire second row by '-3':\[\begin{array}{cccc}1 & 1 & 1 & 2 \0 & 1 & 1 & 1 \0 & 2 & 2 & 2\end{array}\]

Eliminate Above and Below the Second Pivot

Make zeros above and below the '1' in the second row, second column by applying the following operations:1. Replace Row 1 with Row 1 minus Row 2.2. Replace Row 3 with Row 3 minus 2 times Row 2.The updated matrix becomes:\[\begin{array}{cccc}1 & 0 & 0 & 1 \0 & 1 & 1 & 1 \0 & 0 & 0 & 0\end{array}\]

Check for Reduced Row Echelon Form

Ensure every leading entry is '1', the leading entry in each row is the only non-zero entry in its column, and rows of all zeros (if any) are at the bottom of the matrix. The matrix is now in reduced row echelon form.

Key concepts

Reduced Row Echelon Form

The concept of the reduced row echelon form (RREF) is fundamental in solving systems of linear equations. When a matrix is in RREF:

• Each leading entry of a row is 1.
• Each leading 1 is the only non-zero entry in its column.
• The leading 1 in the lower rows must appear to the right of the leading 1s in the rows above.
• Any row of all zeros is placed at the bottom.

This form is incredibly useful because it provides a clear and concise way to interpret the solutions to linear systems. If you have a matrix in RREF, you can immediately tell whether a system has a unique solution, infinitely many solutions, or no solution at all depending on the row structure.

Matrix Transformation

Matrix transformation involves using row operations to change the appearance of a matrix without altering its solution set. The goal with transformations is often to simplify the matrix into an easily interpretable form, such as the reduced row echelon form.
There are three types of elementary row operations used in transformation:

• Swapping two rows.
• Multiplying a row by a non-zero constant.
• Adding or subtracting the multiple of one row to another.

By applying these operations, you can manipulate the matrix while keeping the system of equations it represents intact. This process is strategic and is carefully executed to reveal crucial information about the system, such as dependent variables and consistency.

Linear Algebra

Linear algebra is the branch of mathematics that deals with vectors, vector spaces, and linear equations. At its heart, it is about understanding how linear transformations and spaces work, which is directly related to solving systems of equations.
Key concepts within linear algebra include:

• Vector spaces: Abstract sets that are closed under vector addition and scalar multiplication.
• Matrices: Rectangular arrays of numbers that can represent linear transformations and systems of linear equations.
• Determinants and eigenvalues: Tools for exploring matrix properties and transformations.

Understanding linear algebra is essential for mastering concepts like Gaussian elimination, as it provides the theoretical underpinning that supports these systematic methods of solution. By learning linear algebra, students gain insight into how systems are structured and solved and how equations can represent real-world situations.

Pivot Operations

Pivot operations are central to the process of Gaussian elimination and involve manipulating specific elements within a matrix to simplify it. These operations include making the pivot (the first non-zero entry in a row) equal to 1 and eliminating any other non-zero entries in the pivot's column by row operations.
Steps for pivot operations:

• Identify the pivot in a row. This is usually the first non-zero element from the left.
• Normalize the pivot to 1, often by dividing the entire row by the pivot element itself.
• Use row operations to zero out all other elements in the pivot's column.

Pivot operations step by step reduce the original system of equations, ultimately leading the matrix into reduced row echelon form, where the system can be solved more easily. These operations are crucial because they systematically break down a problem, making it more manageable.