Problem 9 - Precalculus: Mathematical 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. $$\left[\begin{array}{llll} 1 & 2 & 8 & 0 \\ 0 & 1 & 3 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

(a) Yes, it's in row-echelon form. (b) No, not in reduced row-echelon form. (c) System: \(x_1 + 2x_2 + 8x_3 = 0\), \(x_2 + 3x_3 = 2\).

Step 1: Define Row-Echelon Form

A matrix is in row-echelon form if: (1) All zero rows, if any, are at the bottom. (2) Each leading entry of a row is in a column 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: Verify Row-Echelon Form

Examine the given matrix: \[\begin{array}{cccc}1 & 2 & 8 & 0 \0 & 1 & 3 & 2 \0 & 0 & 0 & 0 \end{array}\] - The third row is a zero row and is at the bottom. - The leading entry of the first row is 1, which is to the left of the leading entry in the second row, 1. - There are only zeros below each leading entry. Thus, the matrix is in row-echelon form.

Step 3: Define Reduced Row-Echelon Form

A matrix is in reduced row-echelon form if: (1) It is in row-echelon form. (2) The leading entry in each nonzero row is 1. (3) Each leading 1 is the only nonzero entry in its column.

Step 4: Verify Reduced Row-Echelon Form

Examine the given matrix: \[\begin{array}{cccc}1 & 2 & 8 & 0 \0 & 1 & 3 & 2 \0 & 0 & 0 & 0 \end{array}\] - The matrix is in row-echelon form. - The leading entries are 1, but in the first column, the entries below 1 (first row) and in the same column contain non-zero entries in the second row. Thus, the matrix is not in reduced row-echelon form.

Step 5: Express as System of Equations

The matrix represents an augmented matrix of a system of equations. - The first row represents the equation: \( x_1 + 2x_2 + 8x_3 = 0 \). - The second row represents the equation: \( x_2 + 3x_3 = 2 \). - The third row is a zero row and represents no equation, so no additional information is provided. Thus, the system of equations is:\[\begin{align*}x_1 + 2x_2 + 8x_3 & = 0, \x_2 + 3x_3 & = 2.\end{align*}\]

Key Concepts

System of Linear Equations

A system of linear equations consists of several equations that share the same set of variables. These equations are expressed in terms of unknown variables and constants.
For example, in the context of matrices, each row in an augmented matrix corresponds to a linear equation.
When solving a system of linear equations:

• Our goal is to find the values of the variables that satisfy all the equations simultaneously.
• These systems can be represented and solved using matrices which allow for easier manipulation, especially for larger systems.

In our exercise, the system of equations is derived from the matrix as follows:

• The first row of the matrix, transformed to equation form, is: \[ x_1 + 2x_2 + 8x_3 = 0 \]
• The second row gives: \[ x_2 + 3x_3 = 2 \]
• The third row is all zeros, which doesn't provide any additional equation or constraint.

Understanding these equations helps us interpret what the matrix represents in the context of a real-world problem.

Augmented Matrix

An augmented matrix is a convenient way to write a system of linear equations. It includes both the coefficients of the variables and the constants from the equations' right-hand sides. This form allows us to handle multiple equations simultaneously using matrix operations.
Key characteristics include:

• Each row corresponds to one linear equation.
• The part of the matrix before the vertical line represents the coefficients of the variables.
• The segment after the line contains the constants from each equation.

In the given exercise problem, the augmented matrix is: \[\begin{array}{cccc}1 & 2 & 8 & 0 \0 & 1 & 3 & 2 \0 & 0 & 0 & 0 \\end{array}\] The matrix aligns nicely with our system of equations, showing both the coefficients and the constants clearly.
Using augmented matrices enables us to apply row operations to solve systems in an organized and systematic manner.

Reduced Row-Echelon Form

The reduced row-echelon form (RREF) of a matrix is a specific form that simplifies solving a system of linear equations by making it possible to easily read off the solutions. For a matrix to be in RREF, it must first be in row-echelon form (REF), satisfying additional conditions.

• Each leading entry (the first non-zero entry from the left) in a non-zero row is 1.
• Each leading 1 must be the only non-zero entry in its column. This means every column containing a leading 1 must have all other entries as zeros.

In our example, although the matrix is already in row-echelon form (REF), it is not in reduced row-echelon form because:

• The first row's leading 1 is not the only non-zero entry in its column, as the entries below it should be zeros.

Reaching RREF makes it simple to find unique solutions when they exist. It turns the matrix into a much clearer state, where you can immediately see the value of some of the variables directly or deduce them by simple substitution.