Determine the rank of following matrices by reducing them to echlon form   

          1     2    -1    3

A=     3    -1     2    1

          2    -2     3    2

         1    -1    1    -1

We can determine the rank of matrix \( A \) by reducing it to row echelon form (REF). The rank is the number of non-zero rows in the matrix after it has been reduced.

The matrix \( A \) is given as:

\[
A = \begin{pmatrix}
1 & 2 & -1 & 3 \\
3 & -1 & 2 & 1 \\
2 & -2 & 3 & 2 \\
1 & -1 & 1 & -1
\end{pmatrix}
\]

### Step 1: Use the first row to eliminate the elements below the first pivot (the element in position (1,1)).
Subtract 3 times the first row from the second row, 2 times the first row from the third row, and 1 times the first row from the fourth row.

\[
R_2 = R_2 - 3R_1, \quad R_3 = R_3 - 2R_1, \quad R_4 = R_4 - R_1
\]

This gives:

\[
\begin{pmatrix}
1 & 2 & -1 & 3 \\
0 & -7 & 5 & -8 \\
0 & -6 & 5 & -4 \\
0 & -3 & 2 & -4
\end{pmatrix}
\]

### Step 2: Use the second row to eliminate the elements below the second pivot (the element in position (2,2)).
First, subtract \(\frac{6}{7}\) times the second row from the third row, and \(\frac{3}{7}\) times the second row from the fourth row.

\[
R_3 = R_3 - \frac{6}{7} R_2, \quad R_4 = R_4 - \frac{3}{7} R_2
\]

This gives:

\[
\begin{pmatrix}
1 & 2 & -1 & 3 \\
0 & -7 & 5 & -8 \\
0 & 0 & \frac{1}{7} & \frac{10}{7} \\
0 & 0 & \frac{1}{7} & \frac{2}{7}
\end{pmatrix}
\]

### Step 3: Use the third row to eliminate the elements below the third pivot (the element in position (3,3)).
Subtract the third row from the fourth row.

\[
R_4 = R_4 - R_3
\]

This gives:

\[
\begin{pmatrix}
1 & 2 & -1 & 3 \\
0 & -7 & 5 & -8 \\
0 & 0 & \frac{1}{7} & \frac{10}{7} \\
0 & 0 & 0 & -\frac{8}{7}
\end{pmatrix}
\]

### Step 4: Simplify the pivots.
Multiply the third row by 7 and the fourth row by \(-\frac{7}{8}\) to make the pivots 1:

\[
\begin{pmatrix}
1 & 2 & -1 & 3 \\
0 & -7 & 5 & -8 \\
0 & 0 & 1 & 10 \\
0 & 0 & 0 & 1
\end{pmatrix}
\]

### Step 5: The rank of the matrix
The matrix now has 4 non-zero rows, indicating that all the rows are independent.

**The rank of the matrix is 4.**

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Would you like more details, or do you have any questions? 

Here are five follow-up questions you may find useful:
1. What is the difference between row echelon form (REF) and reduced row echelon form (RREF)?
2. How do elementary row operations affect the determinant of a matrix?
3. Can a matrix have full rank and still be singular?
4. How is the rank of a matrix related to its null space?
5. How can you determine the rank of a matrix using determinants?

**Tip:** The rank of a matrix is crucial in understanding the solution space of a system of linear equations—especially when determining whether a system has unique, infinite, or no solutions.

Determine the rank of the following matrices by reducing them to echelon form.

    1     2   -1    3
A = 3   -1    2    1
      2   -2    3    2
      1   -1    1   -1

Discover how to determine the rank of a 4x4 matrix by reducing it to row echelon form. This detailed solution covers key concepts in linear algebra, including pivot positions, elementary row operations, and the rank theorem. Suitable for undergraduate students learning matrix theory.

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