Problem 7 - 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. $$\left[\begin{array}{rrr}{1} & {0} & {-3} \\ {0} & {1} & {5}\end{array}\right]$$

Step-by-Step Solution

Answer

(a) Yes, in row-echelon form. (b) Yes, in reduced row-echelon form. (c) System: \(x_1 - 3 = 0\), \(x_2 + 5 = 0\).

Step 1: Determine Row-Echelon Form

To determine if a matrix is in row-echelon form, check for the following:1. All non-zero rows are above rows of all zeroes (if any).2. Each leading entry of a row is in a column to the right of the leading entry of the previous row.3. The leading entry (or pivot) in each non-zero row is 1.In the given matrix:\[\begin{bmatrix}1 & 0 & -3 \0 & 1 & 5\end{bmatrix}\]- There are no all-zero rows.- The leading entry of the first row is 1, and the leading entry of the second row (column 2) follows after the first column.Therefore, this matrix is in row-echelon form.

Step 2: Determine Reduced Row-Echelon Form

To determine if a matrix is in reduced row-echelon form, check for additional criteria beyond row-echelon form:1. Every leading 1 is the only non-zero entry in its column.In the matrix:\[\begin{bmatrix}1 & 0 & -3 \0 & 1 & 5\end{bmatrix}\]- The leading 1 in the first row has all other elements in its column as zero.- The leading 1 in the second row has all other elements in its column as zero.Thus, this matrix also satisfies the reduced row-echelon form criteria.

Step 3: Write the System of Equations

For the augmented matrix:\[\begin{bmatrix}1 & 0 & -3 \0 & 1 & 5\end{bmatrix}\]The corresponding system of linear equations is derived by treating each row as an equation where the first two columns represent the coefficients of variables \(x_1\) and \(x_2\), and the third column represents the constants:1. \(x_1 - 3 = 0\)2. \(x_2 + 5 = 0\)

Key Concepts

Reduced Row Echelon Form

The reduced row echelon form is a special kind of matrix where specific rules are applied to simplify matrix equations. It builds upon the conditions of row echelon form by adding stricter criteria:

• Each leading 1 in a row must be the only non-zero entry in that column.
• Each leading 1 appears to the right of the leading 1 in the row above it.

In the provided matrix \(\begin{bmatrix}1 & 0 & -3 \ 0 & 1 & 5\end{bmatrix}\), we can examine these rules:- The leading "1" in the first row is in the first column and has no other non-zero entries within that column.- Similarly, the leading "1" in the second row is in the second column and has no other non-zero entries there.As such, the matrix isn't just in row echelon form; it achieves the status of reduced row echelon form. Reduced row echelon form is powerful because it simplifies solving systems of equations, often directly yielding the variables' values in a straightforward manner.

Augmented Matrix

An augmented matrix is formed by appending the constants from a system of linear equations to the right side of its coefficient matrix. This combines the two elements, allowing for efficient computation and row operations.

The given matrix \(\begin{bmatrix}1 & 0 & -3 \ 0 & 1 & 5\end{bmatrix}\) is already in an augmented format. In this context, the first two columns represent the coefficients for variables \(x_1\) and \(x_2\), while the third column contains constants from the equations.

This augmentation effectively transforms the system of equations into matrix form, aiding upcoming algebraic manipulations. The augmented matrix is a crucial step for techniques like Gaussian elimination, leading towards row echelon or reduced row echelon forms.

System of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. The solution to these equations corresponds to points where all the equations intersect.

In the case of the matrix \(\begin{bmatrix}1 & 0 & -3 \ 0 & 1 & 5\end{bmatrix}\), it represents:- Equation 1: \(x_1 - 3 = 0\)- Equation 2: \(x_2 + 5 = 0\)These equations arise from interpreting each row of the matrix, where the first and second columns are coefficients for \(x_1\) and \(x_2\) respectively, and the third column contains the equations' results.

Solving these provides the values of \(x_1\) and \(x_2\) that satisfy the system. Systems of linear equations are fundamental in algebra, forming the basis of various applications in science, engineering, and beyond, where relationships between multiple variables need to be understood and solved.