Problem 42 - 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. \( \left[\begin{array}{rr} 1 & 3 & 0 & 0 \\ 0 & 0 & 1 & 8 \\ 0 & 0 & 0 & 0 \\ \end{array}\right] \)

Step-by-Step Solution

Answer

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

Step 1: Checking Row-Echelon Form

To inspect if a matrix is in row-echelon form or not, certain conditions have to be met: 1) In each row, the first nonzero number from the left is always to the right of the first nonzero number in the row above it. 2) The bottom rows that have all zero elements if there exist any. Observing the given matrix, it can be seen that these two conditions are satisfied.

Step 2: Checking Reduced Row-Echelon Form

A matrix is in reduced row-echelon form if it satisfies all conditions of row-echelon form and additionally 1) The first nonzero number in each row is 1. 2) The column that contains this 1 has all other elements as zero. Now, inspecting the given matrix again it can be observed that although the first nonzero number in each row is 1, the column that contains this 1 has other elements in it that are not 0 (in this case, the row 2). Hence, our matrix doesn't meet the second condition and is not in reduced row-echelon form.

Key Concepts

Reduced Row-Echelon Form

The reduced row-echelon form (RREF) of a matrix is an essential concept in linear algebra, as it allows for straightforward solutions to systems of linear equations and provides a clear structure to the matrix. A matrix is in RREF if it satisfies all conditions required for row-echelon form (REF) with additional, more specific conditions. These conditions include:

• In each row, the first nonzero number from the left is known as a leading 1.
• The leading 1 in one row should be to the right of any leading 1 in the row above it.
• If a column contains a leading 1, all other elements in that column should be zero.
• Any row of all zeros should be at the bottom of the matrix.

If a matrix adheres to these conditions, it provides a simple and effective way to assess the solutions, often making it easier to find a unique solution or understand the span of the solutions. For example, the matrix provided in the exercise is not in RREF because, despite having leading 1s, the columns of those 1s also contain non-zero elements elsewhere.

Row-Echelon Form Conditions

Row-echelon form (REF) is a simpler and less strict form of organizing the matrix compared to reduced row-echelon form. For a matrix to be in REF, it must satisfy specific conditions that create a simplified triangular form:

• The emerging non-zero leading number in each row is 1, called the leading entry or leading coefficient.
• The leading coefficient of any row must be to the right of the leading coefficient of the row just above it.
• Any rows composed entirely of zeroes are located at the bottom of the matrix.

This form is particularly useful when performing Gaussian elimination to solve systems of linear equations. It allows one to determine the rank of the matrix, helping in understanding the number of solutions possible for a system of equations, whether it's a unique solution, multiple solutions, or none at all. The matrix given in the exercise meets these conditions perfectly, lining up its zeros and ones in a way that fits the REF definition.

Matrix Conditions

Matrices have specific conditions that need to be assessed to understand their forms and properties better. In the context of row-echelon and reduced row-echelon form, these conditions provide a framework for understanding and manipulating the matrix. Here are some general conditions you might look at when assessing a matrix:

• Non-zero rows: In REF and RREF, any row not consisting entirely of zeroes contains a leading entry, ideally a 1 to ease calculations.
• Positions of leading entries: The leading entry of each non-zero row must appear to the right of the leading entry of the row above it for proper echelon forms.
• Zero rows: All-zero rows, if any, must be located at the bottom to maintain structure and simplicity within the matrix.

Understanding these conditions ensures that matrices are properly assessed for solving equations and transformations. It means ensuring that the matrix conveys the correct linear relationships, allowing for accurate interpretation and application in various mathematical scenarios. These standard conditions help keep the structure intuitive for users when transforming a matrix to solve equations or perform further analysis.