Question

Write the matrix in reduced row-echelon form. $$ \left[\begin{array}{rrrr} 1 & 0 & 2 & 1 \\ 0 & 3 & -6 & 6 \\ 2 & 0 & 5 & -4 \\ 0 & 1 & 0 & 1 \end{array}\right] $$

Short answer

The reduced row-echelon form of the given matrix is: \[ \left[\begin{array}{rrrr} 1 & 0 & 2 & 1 \ 0 & 1 & -2 & 2 \ 0 & 0 & 1 & -6 \ 0 & 0 & 0 & 11 \end{array}\right] \]

Step-by-step solution

Identify Pivot Elements

The first pivot element is in the first row, first column. No operation needed.

Scale Second Row

For the ease of calculations, divide the second row by 3: \[ \left[\begin{array}{rrrr} 1 & 0 & 2 & 1 \ 0 & 1 & -2 & 2 \ 2 & 0 & 5 & -4 \ 0 & 1 & 0 & 1 \end{array}\right] \]

Eliminate the Value Below the First Pivot Element

Subtract 2 times the first row from the third row: \[ \left[\begin{array}{rrrr} 1 & 0 & 2 & 1 \ 0 & 1 & -2 & 2 \ 0 & 0 & 1 & -6 \ 0 & 1 & 0 & 1 \end{array}\right] \]

Subtract Second Row From Fourth Row

Subtract the second row from the fourth: \[ \left[\begin{array}{rrrr} 1 & 0 & 2 & 1 \ 0 & 1 & -2 & 2 \ 0 & 0 & 1 & -6 \ 0 & 0 & 2 & -1 \end{array}\right] \]

Final Reduced Row Echelon Form

Subtract 2 times the third row from the fourth row to get to reduced row echelon form: \[ \left[\begin{array}{rrrr} 1 & 0 & 2 & 1 \ 0 & 1 & -2 & 2 \ 0 & 0 & 1 & -6 \ 0 & 0 & 0 & 11 \end{array}\right] \]

Key concepts

Matrix Transformations

A matrix transformation is a process that involves the systematic alteration of a matrix structure by applying specific operations to its rows and columns. In the context of converting a matrix into a reduced row-echelon form, matrix transformations aim to simplify the matrix into a special pattern that is easy to interpret.

The goal is to adjust the matrix such that each leading coefficient (called the pivot) in a row is 1, and all elements below and above the pivot in its column are zeros.

• Think of these transformations as a series of steps that adjust the matrix for clarity and solving systems of linear equations.
• Transformations can involve swapping rows, scaling rows by a non-zero scalar, and adding or subtracting the multiple of one row to another.

Each transformation carries us one step closer to a simpler matrix form, leading eventually to solving linear systems with ease.

Pivot Elements

Pivot elements are crucial for matrix transformations. These are the 'leaders' in the matrix that help guided the process to simplify the matrix step by step. In the context of row-echelon form, each pivot should be 1, and they should ideally appear in a cascading manner from the top-left to the bottom-right of the matrix.

Identifying pivot elements involves:

• Locating the first non-zero element in each row after any row swaps or transformations.
• Ensuring that the pivot position marks the beginning of zero entries in subsequent rows beneath it.

This structure ensures that the reduced row has meaningful clarity and encourages optimal interpretation of the matrix.

Row Operations

Row operations are fundamental techniques used during the matrix transformation process to achieve a reduced row-echelon form. They are simple yet powerful tools that allow us to manipulate the structure of a matrix without changing its solution set, which means the same solutions still apply to the linear system the matrix represents. Here are the three basic row operations:

• Row swapping: interchange two rows to position a better candidate for the pivot.
• Row scaling: multiply or divide a row by a non-zero scalar to change the row's values but not its inherent properties.
• Row addition: add or subtract multiples of one row from another to create a 0 beneath or above a pivot element, working towards the echelon form.

Through these operations, we gradually shape the matrix into its more manageable form, making linear system solutions simpler.

Gaussian Elimination

Gaussian elimination is a methodological approach used to transform a matrix into either row-echelon form or reduced row-echelon form. The primary objective is to use matrix operations we've discussed, like row operations, to systematically reduce the matrix and isolate the pivots in order.

The process of Gaussian elimination involves:

• Iteratively finding pivot elements by eliminating entries below each pivot in successive columns.
• Once reaching reduced row-echelon form, the matrix is left with a staircase-like arrangement that simplifies reading solutions.
• During the entire method, the operations do not affect the system of linear equations the matrix represents, making it an efficient and reliable method for solving matrix problems.

In essence, Gaussian elimination is about uncovering the simplest path to solving equations by reorganizing the data within a matrix.