Problem 41 - Precalculus with Limits

Question

In Exercises 41-44, determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. \( \left[\begin{array}{rr} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 5 \\ 0 & 0 & 0 & 0 \\ \end{array}\right] \)

Step-by-Step Solution

Answer

The given matrix is in row-echelon form, but it is not in reduced row-echelon form.

Step 1: Check if the Matrix is in Row-Echelon Form

A matrix is in row-echelon form if it follows these conditions: (1) Any rows with all zero elements are at the bottom of the matrix. (2) In each non-zero row, the leading (first non-zero) entry is in a column to the right of the leading entry of the row above it. Looking at the given matrix, the rows with all zeroes are at the bottom, and the leading entry of each non-zero row is in a column to the right of the leading entry of the row above it. Therefore the matrix is in row-echelon form.

Step 2: Check if the Matrix is in Reduced Row-Echelon Form

A matrix is in reduced row-echelon form if it meets all the criteria for row-echelon form and two additional conditions: (1) The leading entry in each non-zero row is 1. (2) Each column containing a leading entry has zeroes everywhere else. Looking at the given matrix, the leading entry in each non-zero row is 1. However, in the second column, which contains a leading 1, there is another non-zero integer in the same column. This means the matrix is not in reduced row-echelon form.

Key Concepts

Reduced Row-Echelon Form

Understanding the reduced row-echelon form is crucial in linear algebra as it provides a unique representation for any matrix using elementary row operations. A matrix is said to be in reduced row-echelon form if it fulfills all the conditions of a row-echelon form and additional specific properties.

• The leading entry of each row is 1.
• Every leading 1 is the only non-zero entry in its column.

These additional conditions ensure that reduced row-echelon form provides a standardized way of describing a matrix. This is particularly helpful when solving systems of linear equations, as this form directly highlights the solutions. In the exercise provided, the matrix misses the reduced row-echelon form mark because, although each leading entry is 1, there are non-zero values in the same column as the leading 1 in the second row.

Matrix

A matrix is a rectangular array of numbers or symbols arranged in rows and columns, fundamentally a building block in the study of linear algebra. It is denoted by brackets, as seen in the exercise example.

• Rows are horizontal arrangements, each holding certain values.
• Columns are vertical stacks, aligning values vertically.

Matrices serve as a compact way of representing systems of linear equations, making complex numerical data easier to handle. For instance, in the given exercise, the matrix illustrates a particular system formation that can be visually analyzed to determine its row-echelon or reduced row-echelon status.
Matrices also play a pivotal role in various applications, including computer graphics, statistics, and even machine learning, where they help model and solve intricate problems efficiently.

Linear Algebra

Linear algebra is a branch of mathematics dealing with vectors, vector spaces, and linear transformations. It is essential for understanding how to solve systems of linear equations using methods like matrices and transformations.
One of the most important elements in linear algebra is understanding how systems of equations can be expressed as matrix equations. This abstraction allows mathematicians and scientists to leverage computational techniques for solving large-scale systems efficiently.

• Matrices enable the organization of system coefficients and constants into a structured form.
• Row operations help in simplifying matrices to solve the corresponding system of equations.
• Concepts like row-echelon form simplify matrices, aiding in finding solutions effectively.

In the context of the exercise, determining the form of a matrix using linear algebra concepts guides students in verifying the arrangement rules that apply to matrices in row-echelon and reduced row-echelon form. Overall, mastering linear algebra concepts is essential for progressing in mathematics and applied sciences.