Problem 15 - 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. $$\left[\begin{array}{llll} 1 & 2 & 8 & 0 \\ 0 & 1 & 3 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is in row-echelon form, not in reduced row-echelon form. The system is \(x + 2y + 8z = 0\), \(y + 3z = 2\).

Step 1: Introduction to Row-Echelon Form

A matrix is in row-echelon form if all nonzero rows are above rows of all zeros, and the leading entry of each nonzero row is to the right of the leading entry of the row above it. Also, the leading entry is 1. Identify this pattern in the given matrix.

Step 2: Check for Row-Echelon Form

Examine each row of the matrix. In the given matrix, the first leading entry (1) in row one is to the left of the leading entry (1) in row two. Row three is a zero row. All leading entries above zero rows, and nonzero rows above zero rows, confirm that it is in row-echelon form.

Step 3: Introduction to Reduced Row-Echelon Form

A matrix is in reduced row-echelon form if it is in row-echelon form and all leading entries are 1 and are the only nonzero entries in their column. Verify if the matrix satisfies these conditions.

Step 4: Check for Reduced Row-Echelon Form

In this matrix, while leading entries are 1, the first column contains nonzero entries (2, 8) above the second row's leading 1, violating the reduced row-echelon form rule. Hence, this matrix is not in reduced row-echelon form.

Step 5: Formulate the System of Equations

Translate the non-zero rows into equations. The matrix corresponds to a system of equations: \(1x + 2y + 8z = 0\) and \( y + 3z = 2\). The last row of zeros indicates no further restrictions.

Key Concepts

Row-Echelon Form

Row-echelon form is an important concept when working with matrices and solving systems of linear equations. It helps simplify complex matrices into a form that is easier to work with. To determine if a matrix is in row-echelon form, check three main criteria:

• All nonzero rows (rows with at least one nonzero entry) are placed above any rows of all zeros.
• The leading entry (first nonzero number) in each nonzero row is a 1.
• The leading 1 in each row is to the right of the leading entry in the row just above it.

In the given matrix:\[\begin{bmatrix}1 & 2 & 8 & 0 \0 & 1 & 3 & 2 \0 & 0 & 0 & 0\end{bmatrix}\]the first row has a leading 1 at the beginning, and the second row has its leading 1 further right. Since the zero row is at the bottom, this matrix satisfies all the conditions for row-echelon form.

Reduced Row-Echelon Form

Reduced row-echelon form (RREF) takes the concept of row-echelon form further by imposing additional constraints. A matrix in RREF not only meets all the requirements of row-echelon form but also:

• All leading entries in the matrix must be 1.
• In the columns containing a leading 1, all other entries must be zero.

This results in a matrix where each leading 1 is the sole nonzero entry in its column. Looking at the given matrix again:\[\begin{bmatrix}1 & 2 & 8 & 0 \0 & 1 & 3 & 2 \0 & 0 & 0 & 0\end{bmatrix}\]We can see that while it is in row-echelon form, it is not in reduced row-echelon form. This is because, in the first row, entries 2 and 8 remain nonzero, preventing the matrix from qualifying as reduced row-echelon.

Augmented Matrix

An augmented matrix is a powerful tool in linear algebra used to represent a system of equations. It combines the coefficients of variables and the constants from each equation into a single matrix. This format simplifies complex systems and is essential for performing operations such as row reduction. Consider the following matrix:\[\begin{bmatrix}1 & 2 & 8 & | & 0 \0 & 1 & 3 & | & 2 \0 & 0 & 0 & | & 0\end{bmatrix}\]This becomes the augmented matrix for the system of equations:

• \(1x + 2y + 8z = 0 \)
• \( y + 3z = 2 \)

The vertical line distinguishes the coefficients from the constants on the right.

System of Equations

A system of equations consists of multiple equations that are considered together. These equations typically involve the same set of variables. When an augmented matrix is given, each row corresponds to an equation in the system. Translating the given matrix:\[\begin{bmatrix}1 & 2 & 8 & 0 \0 & 1 & 3 & 2 \0 & 0 & 0 & 0\end{bmatrix}\]We can interpret it as the following system:

• \(1x + 2y + 8z = 0 \)
• \(y + 3z = 2 \)

The third row of zeros indicates no additional equation is necessary. This method provides an organized way of managing linear equations and is essential in solving large systems efficiently. As you work with these systems, understanding their structure can make finding solutions much more manageable.