Problem 20 - Precalculus: Mathematics for Calculus

Question

A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$\left[\begin{array}{llllll} 1 & 3 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 4 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

(a) Yes, the matrix is in REF. (b) No, it's not in RREF. (c) The system is: \( x_1 + 3x_2 + x_4 = 0 \), \( x_2 + 4x_4 = 0 \), \( x_4 + x_5 = 2 \), \( x_4 = 0 \).

Step 1: Understanding Row-Echelon Form (REF)

A matrix is in row-echelon form if it satisfies the following conditions: - All zero rows (if any) are at the bottom of the matrix. - The leading entry (first non-zero element from the left) of each non-zero row is strictly to the right of the leading entry of the row above it. - The leading entry in any non-zero row is 1.

Step 2: Analyze the Matrix for Row-Echelon Form

Review each row: - First leading entry in the first row is 1. - Second row's leading entry is 1 and to the right of the first row's leading entry. - Third row has a leading entry of 1 in the fourth column. - Fourth row is identical to the third row and does not disturb REF, but indicates that an infinite number of solutions exist in a system of equations. - No all-zero rows are present. With these observations, the matrix is indeed in row-echelon form.

Step 3: Understanding Reduced Row-Echelon Form (RREF)

A matrix is in reduced row-echelon form if: - It is in row-echelon form. - The leading entry in each row is the only non-zero entry in its column.

Step 4: Analyze the Matrix for Reduced Row-Echelon Form

Examine the matrix columns: - First column's only non-zero entry is the leading 1 in the first row. - Second column's only non-zero entry is the leading 1 in the second row. - Fourth column has non-zero entries in both the third and fourth rows, violating RREF rules. Thus, the matrix is NOT in reduced row-echelon form.

Step 5: Extracting System of Equations from Matrix

Each row in the matrix represents an equation. Here’s the interpretation:- Row 1: \( x_1 + 3x_2 + x_4 = 0 \)- Row 2: \( x_2 + 4x_4 = 0 \)- Row 3: \( x_4 + x_5 = 2 \)- Row 4: \( x_4 = 0 \)

Step 6: Conclusion

Based on our analysis, the matrix is in row-echelon form but not in reduced row-echelon form. The extracted system of equations from the matrix describes the relationships between the variables as derived in Step 5.

Key Concepts

Reduced Row-Echelon Form

Understanding the concept of Reduced Row-Echelon Form (RREF) is crucial when working with matrices. A matrix in RREF not only satisfies the conditions of Row-Echelon Form (REF), but it also meets additional criteria:

• The leading entry in each row is the only non-zero entry in its column.
• This means that each leading 1 must "stand alone" in its column, with all other entries in that column being zero.

RREF is often the goal of matrix transformations since it simplifies solving systems of linear equations. When a matrix is in RREF, it's much easier to identify solutions to the corresponding system of equations. However, getting a matrix into this form requires careful manipulation of rows, often using operations like row addition, scalar multiplication, or swapping rows. In our case, the given matrix doesn’t comply with all RREF conditions because the fourth column has non-zero entries in the third and fourth rows. Thus, while it is in REF, it is not in RREF.

System of Linear Equations

When analyzing a matrix like the given one, each row can be interpreted as an equation in a system of linear equations. The process of converting a matrix into a system of equations involves:

• Assigning a variable to each column that corresponds to a coefficient, often starting with \( x_1, x_2, x_3, \ldots \).
• The rightmost column, if present, typically represents constants or results of the equations.

For the given matrix, the first row translates to the equation \( x_1 + 3x_2 + x_4 = 0 \). Each row should be interpreted in this way to understand how they define relationships between the variables. This method offers a visualization of the problems and constraints inherent in the system.

Matrix Analysis

Matrix Analysis is a field of study focused on understanding the structure and properties of matrices. When analyzing a matrix like the one provided, it is essential to identify its form, such as whether it is in row-echelon form or reduced row-echelon form.

• Analyzing includes determining the potential solutions of a system of equations that the matrix can represent.
• One investigates properties like the presence of zero rows, leading coefficients, and their positions.

By examining the matrix's rows and columns, we can deduce the nature of the solutions. Sometimes, matrices reveal multiple or infinite solutions, especially when rows are identical or imply dependency among equations, as indicated in our task. Matrix Analysis involves techniques that help understand how to manipulate matrices to find solutions effectively, such as Gaussian elimination and matrix transformations.