The matrix given is a matrix from a sequence of matrices obtained when reducing a matrix to echelon form. The next step in the sequence reduces another column to the appropriate form. Find the next step in the sequence:                                   1 0 3 4

0 1 4 −3

0 0 2 8

The given matrix appears to be in row echelon form, and the task is to further reduce it. The matrix provided is:

\[
\begin{pmatrix}
1 & 0 & 3 & 4 \\
0 & 1 & 4 & -3 \\
0 & 0 & 2 & 8
\end{pmatrix}
\]

The next step is to convert this matrix to reduced row echelon form (RREF). To do this, we will make the leading coefficient of the third row a 1 and eliminate the third column's coefficients in the rows above it.

### Step 1: Make the Leading Coefficient in the Third Row a 1

To make the leading coefficient in the third row a 1, divide the entire third row by 2:

\[
\begin{pmatrix}
1 & 0 & 3 & 4 \\
0 & 1 & 4 & -3 \\
0 & 0 & 1 & 4
\end{pmatrix}
\]

### Step 2: Eliminate the Third Column's Coefficients Above the Leading 1

To eliminate the 3 above the leading 1 in the third row:

- **Row 1:** Subtract 3 times the third row from the first row.
- **Row 2:** Subtract 4 times the third row from the second row.

#### Row Operations:

- **Row 1 operation:** \( R_1 \rightarrow R_1 - 3R_3 \)

\[
\begin{pmatrix}
1 & 0 & 3 & 4 \\
0 & 1 & 4 & -3 \\
0 & 0 & 1 & 4
\end{pmatrix}
\rightarrow
\begin{pmatrix}
1 & 0 & 0 & -8 \\
0 & 1 & 4 & -3 \\
0 & 0 & 1 & 4
\end{pmatrix}
\]

- **Row 2 operation:** \( R_2 \rightarrow R_2 - 4R_3 \)

\[
\begin{pmatrix}
1 & 0 & 0 & -8 \\
0 & 1 & 4 & -3 \\
0 & 0 & 1 & 4
\end{pmatrix}
\rightarrow
\begin{pmatrix}
1 & 0 & 0 & -8 \\
0 & 1 & 0 & -19 \\
0 & 0 & 1 & 4
\end{pmatrix}
\]

The matrix is now in reduced row echelon form (RREF). This form ensures that each leading coefficient is 1, and all entries above and below the leading coefficients are 0.

Would you like more details or have any questions?

Here are some related questions:
1. What is the difference between row echelon form and reduced row echelon form?
2. How can the RREF of a matrix be used to solve a system of linear equations?
3. What is the significance of pivot positions in the context of RREF?
4. How does row reduction affect the determinant of a matrix?
5. What role do elementary row operations play in matrix transformations?
6. Can a matrix have more than one RREF? Why or why not?
7. How does one determine the rank of a matrix using its RREF?
8. How are free variables identified in the RREF of a matrix?

**Tip:** When performing row operations, always double-check each step to ensure that each element is correctly calculated. This helps prevent errors in reaching the correct final form.

Explore the next step in reducing a matrix to echelon form, focusing on converting the given matrix to reduced row echelon form (RREF). This detailed solution covers matrix operations, row echelon form, and specific steps to achieve RREF. Suitable for undergraduate students and learners interested in linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation