Start 4 By 4 Table 1st Row 1st Column negative 21 2nd Column negative 6 3rd Column 15 4st Column negative 33 2nd Row 1st Column 10 2nd Column 6 3rd Column negative 8 4st Column 20 3rd Row 1st Column negative 6 2nd Column negative 10 3rd Column 2 4st Column negative 9 4st Row 1st Column 7 2nd Column negative 9 3rd Column negative 2 4st Column negative 4 EndTable 
 
what is the reduced echelon form

To find the reduced row echelon form (RREF) of a matrix, we need to perform Gaussian elimination followed by back substitution to make each leading coefficient a "1" (a pivot) and ensure that all other entries in the column of the pivot are zeros.

Let's first represent the 4x4 matrix from the provided table:

\[
\textbf{A} = 
\begin{bmatrix}
-21 & -6 & 15 & -33 \\
10 & 6 & -8 & 20 \\
-6 & -10 & 2 & -9 \\
7 & -9 & -2 & -4
\end{bmatrix}
\]

Now, let's perform the row reduction steps to get the matrix into reduced echelon form.The reduced row echelon form (RREF) of the given 4x4 matrix is:

\[
\text{RREF}(\textbf{A}) = 
\begin{bmatrix}
1 & 0 & 0 & -\frac{23}{50} \\
0 & 1 & 0 & \frac{33}{50} \\
0 & 0 & 1 & -\frac{129}{50} \\
0 & 0 & 0 & 0
\end{bmatrix}
\]

This matrix is in reduced echelon form where each leading entry is 1, and all other entries in the columns containing leading entries are zeros.

Would you like more details on how the row reduction was performed or have any other questions?

Here are 5 related questions you might find interesting:
1. How is Gaussian elimination different from Gauss-Jordan elimination?
2. What is the significance of the reduced echelon form in solving linear systems?
3. How do you find the rank of a matrix from its RREF?
4. What happens if a matrix is not invertible? How does it affect the RREF?
5. How do you determine if a system of linear equations has a unique solution, infinitely many solutions, or no solution?

**Tip:** Understanding the process of row reduction is crucial for solving systems of linear equations efficiently!

Start 4 By 4 Table 1st Row 1st Column negative 21 2nd Column negative 6 3rd Column 15 4st Column negative 33 2nd Row 1st Column 10 2nd Column 6 3rd Column negative 8 4st Column 20 3rd Row 1st Column negative 6 2nd Column negative 10 3rd Column 2 4st Column negative 9 4st Row 1st Column 7 2nd Column negative 9 3rd Column negative 2 4st Column negative 4 EndTable. What is the reduced echelon form?

Learn how to compute the reduced row echelon form (RREF) of a 4x4 matrix using Gaussian elimination. This problem walks through the key steps in transforming a matrix into RREF and explains important concepts in linear algebra, such as pivot elements and back substitution. Suitable for Grades 10-12 or introductory college students.

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