Problem 49 - Precalculus with Limits

Question

In Exercises 49-54, use the matrix capabilities of a graphing utility to write the \(matrix\) in reduced row-echelon form. \( \left[\begin{array}{rrr} 3 & 3 & 3 \\ -1 & 0 & -4 \\ 2 & 4 & -4 \\ \end{array}\right] \)

Step-by-Step Solution

Answer

The reduced row-echelon form of the given matrix is \[ \begin{array}{rrr} 1 & 1 & 1 \ 0 & 0 & 0 \ 0 & 1 & -3 \ \end{array} \]

Step 1: Multiply row 1 by \( \frac{1}{3} \)

This operation is done to make the leading coefficient of row 1 as 1. Which gives us the new matrix: \[ \begin{array}{rrr} 1 & 1 & 1 \ -1 & 0 & -4 \ 2 & 4 & -4 \ \end{array} \]

Step 2: Add row 1 to row 2

This is done to get 0 as leading coefficient of row 2. The matrix now becomes: \[ \begin{array}{rrr} 1 & 1 & 1 \ 0 & 1 & -3 \ 2 & 4 & -4 \ \end{array} \]

Step 3: Substitute row 3 with row 3 - (2*row 1)

This step is performed to get 0 as leading coefficient of row 3. The matrix now becomes: \[ \begin{array}{rrr} 1 & 1 & 1 \ 0 & 1 & -3 \ 0 & 2 & -6 \ \end{array} \]

Step 4: Substitute row 3 with row 3 - (2*row 2)

This step makes the leading coefficient of row 3 as 0: \[ \begin{array}{rrr} 1 & 1 & 1 \ 0 & 1 & -3 \ 0 & 0 & 0 \ \end{array} \]

Step 5: Swap row 2 and row 3

This is the final step to achieve the reduced row-echelon form. We get: \[ \begin{array}{rrr} 1 & 1 & 1 \ 0 & 0 & 0 \ 0 & 1 & -3 \ \end{array} \]

Key Concepts

Matrix Operations

Matrix operations are the building blocks of many problems in linear algebra. They involve various techniques that allow us to manipulate matrices to solve equations, transform data, or even simplify expressions.

• Addition and Subtraction: You add or subtract matrices by adding or subtracting their corresponding elements.
• Scalar Multiplication: You multiply a matrix by a scalar (a real number) by multiplying every entry of the matrix by that scalar.
• Matrix Multiplication: This involves the dot product of rows and columns between two matrices.

Performing these operations can transform a complicated matrix into one that's easier to understand and work with, such as when finding its reduced row-echelon form.

Gaussian Elimination

Gaussian elimination is a methodical process used to convert a matrix into its row-echelon form. This process involves a series of "elementary row operations" to simplify a system of linear equations.
The method involves three types of operations:

• Swap: Exchanging two rows.
• Scale: Multiply a row by a non-zero scalar.
• Add: Add or subtract a multiple of one row to another.

The goal is to make the matrix easier to interpret and to find consistent solutions to linear equations. By following this procedure systematically, we ensure that all possible transformations are accounted for and valid.

Linear Algebra

Linear algebra is a branch of mathematics focused on vectors, matrices, and linear transformations. It's a framework that allows us to understand and solve linear equations, which are foundational in many fields of science and engineering.
Some vital concepts include:

• Vectors: Objects that have both magnitude and direction.
• Matrices: Rectangular arrays of numbers, which can represent systems of linear equations.
• Systems of Equations: Sets of equations with multiple variables that can be represented and solved using matrices.

Linear algebra provides the tools needed to solve real-world problems efficiently, whether it's analyzing data or modeling complex physical systems.

Matrix Row Reduction

Matrix row reduction is a step-by-step approach in transforming a matrix into its reduced row-echelon form (RREF). This form makes it straightforward to read off solutions to a system of equations or to determine the rank of a matrix.
The goal of matrix row reduction is to:

• Make the leading coefficient (the first non-zero number from the left, also called the pivot) in each row equal to one.
• Ensure that each pivot is the only non-zero entry in its column.
• Transform any matrix row that does not have a pivot into a zero row.

This process not only simplifies solving systems of equations but also helps in understanding the structure and properties of the matrix itself.