Problem 53 - 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. \( \left[\begin{array}{rrr} -3 && 5 && 1 && 12 \\ 1 && -1 && 1 && 4 \\ \end{array}\right] \)

Step-by-Step Solution

Answer

The reduced row-echelon form of the given matrix is \[\left[\begin{array}{rrr} 1 && 0 && 1 && 4 \ 0 && 1 && 2 && 12 \ \end{array}\right]\].

Step 1: Setup

Start with the original matrix: \[ \left[\begin{array}{rrr} -3 && 5 && 1 && 12 \ 1 && -1 && 1 && 4 \ \end{array}\right] \]

Step 2: First Row Operation

Multiply the first row by -1/3 to make the leading entry in the first row '1': \[\left[\begin{array}{rrr} 1 && -5/3 && -1/3 && -4 \ 1 && -1 && 1 && 4 \ \end{array}\right]\]

Step 3: Second Row Operation

Subtract the first row from the second to zero out the first entry in the second row: \[\left[\begin{array}{rrr} 1 && -5/3 && -1/3 && -4 \ 0 && 2/3 && 4/3 && 8 \ \end{array}\right]\]

Step 4: Third Row Operation

Multiply the second row by 3/2 to get the leading '1' in the second row: \[\left[\begin{array}{rrr} 1 && -5/3 && -1/3 && -4 \ 0 && 1 && 2 && 12 \ \end{array}\right]\]

Step 5: Final Row Operation

Add 5/3 times the second row to the first to zero out the off-diagonal entries: \[\left[\begin{array}{rrr} 1 && 0 && 1 && 4 \ 0 && 1 && 2 && 12 \ \end{array}\right]\] This is the reduced row-echelon form of the original matrix.

Key Concepts

Matrix Operations

Matrix operations are fundamental tools in linear algebra, allowing us to perform calculations and transformations on matrices. These operations include addition, subtraction, scalar multiplication, and, importantly, row operations, which are crucial for simplifying matrices. A key goal in matrix operations is often to transform a given matrix into a simpler form, such as reduced row-echelon form (RREF). This form makes it easier to understand and solve systems of linear equations. Basic matrix operations:

• **Addition/Subtraction**: Matrices of the same dimensions can be added or subtracted element-wise.
• **Scalar Multiplication**: Each element of a matrix is multiplied by a constant (scalar).
• **Row Operations**: Specific operations changing rows to reach an RREF.

These operations play a pivotal role in performing and understanding many linear transformations and solutions in linear algebra.

Graphing Utility

A graphing utility is a technological tool that helps visualize and compute mathematical problems, including matrices. These can be either physical calculators often used in academic settings or software tools available on computers and mobile devices. *Benefits of Using Graphing Utilities:*

• They simplify complex matrix operations by automating calculations.
• Provide visual insights through graphing capabilities, which are especially beneficial when dealing with systems of equations.
• Can check manual computations quickly to ensure accuracy.

To use graphing utilities for matrices like converting to RREF: 1. Input the matrix into the graphing tool. 2. Use functions to apply row operations automatically. 3. Analyze the output to verify the solution. This ability makes graphing utilities a valuable aid to anyone studying linear algebra, as they allow users to focus on understanding concepts rather than manual computation.

Row Operations

Row operations are crucial techniques used to transform matrices into simpler forms, like the reduced row-echelon form (RREF), which then make the solutions to systems of equations easier to deduce. There are three types of elementary row operations: 1. **Row Addition/Subtraction:** Adding or subtracting the multiples of one row from another. This operation helps eliminate variables systematically. 2. **Row Multiplication:** Multiplying all entries of a row by a non-zero scalar, normally done to obtain leading 1's in a row. 3. **Row Interchange:** Swapping two rows when necessary to make computation simpler. These operations are applied carefully with the aim of simplifying a matrix while preserving its fundamental properties. For instance, solving the given matrix involved these operations to reach the RREF which makes identifying the solutions to the system of equations quite direct.

Linear Algebra

Linear algebra is a branch of mathematics that focuses on vector spaces, linear mappings, and systems of linear equations. It is the foundation of many areas of modern mathematics and has extensive applications in areas like physics, engineering, computer science, and more. *Key Concepts in Linear Algebra Include:*

• **Matrices and Determinants:** Essential tools for describing linear systems and transformations.
• **Vector Spaces:** Collections of vectors that can be scaled and added together.
• **Linear Transformations:** Functions that map vector spaces to themselves or other vector spaces.

In the context of linear algebra, reducing a matrix to its reduced row-echelon form is a method to solve linear equations. It also provides insights into the solutions’ nature, such as existence and uniqueness, making it an integral process in many mathematical and applied problems.