Problem 45 - 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. $$\left[\begin{array}{rrr} 3 & 3 & 3 \\ -1 & 0 & -4 \\ 2 & 4 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

The reduced row-echelon form of the given matrix is: \[ \left[\begin{array}{rrr} 1 & 0 & -2 \ 0 & 1 & -3 \ 0 & 0 & 1 \end{array}\right] \]

Step 1: Identify the matrix

The first thing to do is to identify the matrix that we are working with. This is the matrix we are given: \[ \left[\begin{array}{rrr} 3 & 3 & 3 \ -1 & 0 & -4 \ 2 & 4 & -2 \end{array}\right] \] This is a 3x3 matrix, meaning it has three rows and three columns.

Step 2: Begin row operations

We start by replacing R1 by 1/3*R1, R2 by -1*R2 and R3 remains the same. This results in the following matrix: \[ \left[\begin{array}{rrr} 1 & 1 & 1 \ 1 & 0 & 4 \ 2 & 4 & -2 \end{array}\right] \]

Step 3: Continue row operations

In this step, let's replace R2 by R2 - R1 and R3 by R3 - 2*R1 to make sure the elements below the first element in the first column (leading 1) are zeros. This will give us: \[ \left[\begin{array}{rrr} 1 & 1 & 1 \ 0 & -1 & 3 \ 0 & 2 & -4 \end{array}\right] \]

Step 4: Further row operations

Next, we replace R2 by -1*R2 and R3 by R3 - 2*R2 in order to make the second element in second column of first row and all the elements below the leading 1 in the second row zeros. As a result, we get: \[ \left[\begin{array}{rrr} 1 & 0 & -2 \ 0 & 1 & -3 \ 0 & 0 & 2 \end{array}\right] \]

Step 5: Final row operation

Lastly, we need to make the third element in third row a 1 by replacing R3 by 1/2*R3, so we end up with the reduced row-echelon form: \[ \left[\begin{array}{rrr} 1 & 0 & -2 \ 0 & 1 & -3 \ 0 & 0 & 1 \end{array}\right] \] This is the final matrix in reduced row-echelon form.

Key Concepts

Matrix Row Operations

Understanding matrix row operations is essential for solving systems of equations and transforming matrices into their reduced row-echelon form (RREF). These operations include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of rows from one another.

During the process of achieving RREF, the goal is to obtain a leading 1 in each row, positioned to the right of the leading 1 in the row above, if any. All elements below and above each leading 1 are transformed to zero using these operations. This systematic approach facilitates the solving of linear equations represented by the matrix.

For the exercise provided, the row operations begin with normalizing the first row to create a leading 1 and then proceeding to eliminate other numbers in the first column. Subsequent steps involve clearing out the entries above and below the leading ones in each row, achieving columnar zeros except for the diagonal of ones. Finally, the last row is scaled to ensure the last variable corresponds directly to its coefficient.

3x3 Matrix

A 3x3 matrix consists of 3 rows and 3 columns, forming a square array of nine numbers. In mathematics, specifically in linear algebra, such matrices are commonly used to represent systems of linear equations or to perform transformations in three-dimensional space.

When working with a 3x3 matrix, one can determine its determinant, eigenvalues, and eigenvectors, which are significant in various applications, including physics, engineering, and computer graphics. The process of row reduction to achieve RREF, as demonstrated in the given exercise, is particularly valuable as it simplifies the matrix, often leading to clearer insights into the system's behavior or solutions.

In instructional settings, 3x3 matrices are frequently used due to their complexity, which is enough to demonstrate most concepts in linear algebra without being too unwieldy. The example in the exercise illustrates the transformation into RREF, which is a critical step in solving linear systems or inverting the matrix, crucial for understanding how systems of equations interact.

Graphing Utility

A graphing utility is a powerful tool often used in mathematics education to facilitate the understanding of complex concepts. It can visually represent equations, plot functions, and perform various computations, including matrix operations. In the context of linear algebra, a graphing utility could be used to quickly compute the RREF of matrices, as suggested in the exercise.

Modern graphing utilities, which may come as standalone devices or software applications, often contain functionalities for entering matrices and performing row operations automatically. This is particularly helpful for students to check their manual work and to grasp the impact of individual row operations. It's noteworthy that while these utilities are excellent for verification and visualization, the manual process, as demonstrated in the provided step by step solution, is crucial for learning and understanding the underlying principles of matrix manipulations.

By using a graphing utility, students can spend less time on arithmetic calculations and more time analyzing the results and interpreting the meanings of the transformed matrices, thus enhancing their comprehension of concepts in algebra and beyond.