Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form . Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form. $$\left[\begin{array}{ccc} 1 & 3 & 2 \\ 5 & 15 & 9 \\ 2 & 6 & 10 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix in reduced row-echelon form is \( \left[\begin{array}{ccc} 1 & 3 & 0 \ 0 & 0 & 1 \ 0 & 0 & 0 \end{array}\right] \).

Step 1 - Identify Leading Entry

For the first row, the leading entry is 1, and it's already 1, so no action is required for the first row. This meets the criteria of having a leading 1.

Step 2 - Make Zeroes Underneath The Leading Entry

For the second row, subtract 5 times the first row from the second row - it will make the first term zero. Similarly, for the third row, subtract 2 times the first row from the third row. Now the matrix becomes: \( \left[\begin{array}{ccc} 1 & 3 & 2 \ 0 & 0 & -1 \ 0 & 0 & 6 \end{array}\right] \).

Step 3 - Identify New Leading Entry and Normalize

For the second row, the leading entry becomes -1. Multiply the whole second row by -1 to make it positive 1 (it's the second criteria of RREF to have positive leading entries). For the third row, divide the whole row by 6 making 1 as leading entry. The matrix becomes : \( \left[\begin{array}{ccc} 1 & 3 & 2 \ 0 & 0 & 1 \ 0 & 0 & 1 \end{array}\right] \)

Step 4 - Make Zeroes Above Leading Entries

To achieve the third RREF criteria, subtract the second row from the first row. Also subtract the third row from the first row. The matrix becomes now : \( \left[\begin{array}{ccc} 1 & 3 & 0 \ 0 & 0 & 1 \ 0 & 0 & 0 \end{array}\right] \)

Key Concepts

Matrix Transformations

Matrix transformations involve changing a matrix into another form through systematic operations. Such transformations are essential in turning a matrix into a more usable format, like the reduced row-echelon form (RREF). The RREF of a matrix is a simplified version that makes it straightforward to solve systems of linear equations or analyze the properties of the matrix.

Through matrix transformations, you can accomplish many things:

• Clear understanding of linear independence.
• Simple determination of the rank and determinant.
• Efficiency in solving linear systems.

Applying sequence operations to achieve this includes swapping rows, multiplying rows by non-zero scalars, and adding or subtracting the multiples of other rows from a given row. Each step in this process gets you closer to the RREF, where every leading entry (the first non-zero number from the left in a row) is 1, and all entries below and above these leading 1s are zeroes.

Graphing Utility

A graphing utility is an application or calculator that lets you perform advanced mathematical operations and visualize the results. It includes matrix operations like finding inverses, determinants, and turning matrices into RREF. Such tools save time and help in verifying calculations that are cumbersome by hand.

Using a graphing utility for matrix transformations provides:

• Quick computation and visualization of RREF.
• Verification of results obtained manually.
• A better understanding of each transformation step.

By ensuring accuracy and providing visual context, these tools support deeper insights into the matrix operations and make learning and applying mathematical concepts more intuitive.

Leading Entry

In matrix terms, a leading entry is the first non-zero number in a row from the left. It's critical in the process of transforming a matrix into its reduced row-echelon form. The leading entry is ideally a 1 in RREF, and you make sure that all numbers below and above it are zeroes. This characteristic makes it easy to interpret the matrix regarding solutions to linear systems.

The importance of leading entries involves:

• Helping identify pivot positions that are key for solving equations.
• Aiding the calculation of the rank of the matrix.
• Determining redundancy or independence of linear equations.

By manipulating leading entries across steps of transformation, matrices become more organized, making the matrix intuitive to both compute and comprehend.

Matrix Row Operations

Matrix row operations are the fundamental building blocks for matrix transformations. These operations include:

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

Each operation must be applied carefully to preserve the solutions of the corresponding linear equations. These transformations are structured to simplify the matrix while maintaining the equivalent system of equations.

Competence with row operations brings several more benefits:

• Breaking down complex equations into simpler tasks.
• Insight into the properties of matrixes such as determinants and ranks.
• The foundation for more advanced linear algebra operations.

Whether you're working with manual calculations or employing a graphing utility, mastering matrix row operations is key to proficient problem solving and mathematical reasoning in linear algebra.