Problem 11 - 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. $$\left[\begin{array}{llll}{1} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} \\\ {0} & {1} & {5} & {1}\end{array}\right]$$

Step-by-Step Solution

Answer

(a) Yes, it is in row-echelon form. (b) No, it is not in reduced row-echelon form. (c) \(x_1 = 0\), \(x_2 + 5x_3 = 1\).

Step 1: Understand Row-Echelon Form

To check if a matrix is in row-echelon form, ensure that:- All nonzero rows are above any rows of all zeros.- Each leading entry of a row is in a column to the right of the leading entry of the row above it.- The leading entry in any nonzero row is 1.For the given matrix:\[\begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 1 & 5 & 1 \end{bmatrix}\]1st nonzero row's leading entry: 1 (column 1)2nd nonzero row's leading entry: 1 (column 2)The matrix satisfies the conditions for row-echelon form.

Step 2: Check for Reduced Row-Echelon Form

In reduced row-echelon form, in addition to being in row-echelon form:- Every leading entry is the only nonzero entry in its column.For the matrix:\[\begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 1 & 5 & 1 \end{bmatrix}\]The leading 1s are in the columns 1 and 2. Column 2 also contains 5 besides the leading 1 in the third row, making it not reduced row-echelon form.

Step 3: Write the System of Equations

An augmented matrix represents a system of equations. For matrix form:\[\begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 1 & 5 & 1 \end{bmatrix}\]It implies:1. \(x_1 = 0\)2. No equation from the second row as it's all zeros.3. \(x_2 + 5x_3 = 1\)The system is:1. \( x_1 = 0 \)2. \( x_2 + 5x_3 = 1 \)

Key Concepts

System of Equations

When we talk about a system of equations, we're referring to a set of equations that are solved together. Each equation in the system corresponds to a row in the matrix. The variables in the equations are usually represented by the columns in the matrix. In the context of an augmented matrix, one or more rows show how different variables relate to each other and can be used to find solutions.

In our exercise, the augmented matrix translates into these equations:

• The first row gives us: \( x_1 = 0 \).
• The second row is all zeros, indicating no new equation.
• The third row corresponds to: \( x_2 + 5x_3 = 1 \).

Augmented matrices efficiently represent multiple equations in one concise format, making it easier to manipulate the entire system instead of solving each equation separately. This method is helpful in many mathematical and real-world applications, such as physics or engineering problems.

Reduced Row Echelon Form

Reduced Row Echelon Form, often abbreviated as RREF, is a form of a matrix providing a standardized way of viewing the solution of a system of equations. It takes row-echelon form one step further by ensuring that each leading entry (the first non-zero number from the left in a row) is the only non-zero number in its column.

Considering the given matrix:

• The matrix is in row-echelon form since each leading entry is 1, and the leading entries increase from left to right as you move down the rows.
• Nevertheless, the matrix is not in RREF because the column containing a leading entry (in this case, column 2) should only have that one non-zero entry, but here it also has a "5" which violates the RREF rule.

This specific identification helps during the solution process since a matrix in RREF allows us to directly "read off" the solutions to the systems of equations it represents. It is the ultimate stage of simplification of a matrix and makes identifying specific variable solutions straightforward.

Leading Entry

In matrix terminology, the leading entry of a row is the first non-zero number you encounter when moving from left to right. This concept is crucial as it helps determine the matrix's form and whether it adheres to specific matrix properties like row-echelon form or reduced row-echelon form.

• In row-echelon form, the leading entry of each row appears to the right of the leading entry in the previous row.
• Leading entries in our matrix example are "1" in the first and third rows, indicating the matrix follows the row-echelon form criteria.

The identification of leading entries is vital because they guide us in performing operations, like row swaps or scalar multiplications, that simplify the system of equations and help to reach the RREF.