Problem 10 - Algebra and Trigonometry

Question

7–14 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. $$\left[\begin{array}{rrrr}{1} & {0} & {-7} & {0} \\ {0} & {1} & {3} & {0} \\\ {0} & {0} & {0} & {1}\end{array}\right]$$

Step-by-Step Solution

Answer

(a) Yes, the matrix is in row-echelon form. (b) No, it is not in reduced row-echelon form. (c) The system is inconsistent: \(x - 7z = 0\), \(y + 3z = 0\), \(0 = 1\).

Step 1: Identify row-echelon form (REF) criteria

To be in row-echelon form, a matrix must meet the following conditions: 1. All nonzero rows appear above any rows of all zeros. 2. Each leading entry of a row is strictly to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zero.

Step 2: Check matrix against REF criteria

Examine the given matrix:\[\begin{bmatrix}1 & 0 & -7 & 0 \0 & 1 & 3 & 0 \0 & 0 & 0 & 1\end{bmatrix}\]- The first condition is satisfied as there are no zero rows above nonzero rows.- The leading entry in each row appears to the right of the leading entry in the row above it.- All entries below each leading entry are zero. Thus, the matrix is in row-echelon form.

Step 3: Identify reduced row-echelon form (RREF) criteria

To be in reduced row-echelon form, a matrix must meet all REF criteria, and additionally: 1. Every leading entry is 1. 2. Each leading 1 is the only nonzero entry in its column.

Step 4: Check matrix against RREF criteria

Examine the given matrix again:\[\begin{bmatrix}1 & 0 & -7 & 0 \0 & 1 & 3 & 0 \0 & 0 & 0 & 1\end{bmatrix}\]- Every leading entry is 1.- However, the second column has an additional nonzero entry (3) in the first row, meaning column 3 has a nonzero entry (-7) not associated with a leading entry 1. Thus, it does not satisfy the criteria for reduced row-echelon form.

Step 5: Write the system of equations from the augmented matrix

To express the matrix as a system of equations, treat each row as an equation. The matrix is augmented, so the last column will be the constants on the right side of the equations.1. First row: \( 1x + 0y - 7z = 0 \) simplifies to \( x - 7z = 0 \).2. Second row: \( 0x + 1y + 3z = 0 \) simplifies to \( y + 3z = 0 \).3. Third row: \( 0x + 0y + 0z = 1 \) which is inconsistent. In general practice, you would recognize this indicates no solution.

Key Concepts

Row-Echelon Form

The row-echelon form (REF) of a matrix is a special configuration that simplifies solving systems of linear equations. This form is characterized by a unique arrangement where each row has a leading entry (first non-zero number) further to the right than the row above it. Let's break it down:

• Non-zero rows are arranged above any rows that are completely zeroes.
• The leading entry of each row is strictly to the right of the leading entry of the row above.
• All entries in each column below a leading entry must be zero.

Understanding these criteria can help you identify if a matrix is in row-echelon form and ensures that solving the corresponding system of equations is as simple as possible.

Reduced Row-Echelon Form

Reduced row-echelon form (RREF) takes the conditions of row-echelon form and adds a few more to refine a matrix further. A matrix in RREF is even more methodically structured:

• Every leading entry is exactly 1, ensuring simplicity and ease of interpretation.
• Each leading 1 is the sole non-zero entry in its column, eliminating unnecessary entries for clearer solutions.

RREF is an essential form in linear algebra because it quickly shows solutions to systems of equations or highlights inconsistencies such as no solution or infinitely many solutions. In our matrix, while every leading entry was 1, we found additional non-zero entries in the rows that hindered it from meeting RREF requirements.

Augmented Matrix

An augmented matrix is a powerful tool that represents a system of linear equations. It includes the coefficients of the variables and the constants from the equations. Visually, you'll notice:

• The columns before the last contain the coefficients of the variables in the equations.
• The last column holds the constants, which would usually appear on the right side of the equal sign in standard equations.

An augmented matrix simplifies the visualization and manipulation of systems, allowing you to focus on operations that transform the system into desired forms like the row-echelon or reduced row-echelon forms.

System of Equations

A system of equations consists of multiple equations that are solved together for a common solution. By representing it with matrices, the complex relationships in multiple variables become easier to handle using linear algebra techniques. Here's what you need to know:

• Each row of the matrix represents a separate linear equation.
• By converting a matrix back into a system, you can interpret the solution, identify relationships among variables, or detect inconsistencies.

In our matrix example, the system derived had one inconsistency, indicating no solutions exist. Consistently analyzing each row is key to understanding the situation of the system, whether it might have a single solution, many, or none at all.