-1 - 3 3 - 1

1 1 - 1 0

2 - 5 2 - 3

-1 1 0 1

Reduce the matrix into echelon form and find the rank of matrix

To reduce the given matrix into echelon form and determine its rank, let's follow these steps:

### Given Matrix
\[
\begin{pmatrix}
-1 & -3 & 3 & -1 \\
1 & 1 & -1 & 0 \\
2 & -5 & 2 & -3 \\
-1 & 1 & 0 & 1
\end{pmatrix}
\]

### Step 1: Row Operations to Get Echelon Form

#### 1. First, let's make the first element of the first row (pivot) positive by swapping Row 1 with Row 2.

\[
\text{Swap } R_1 \text{ and } R_2:
\begin{pmatrix}
1 & 1 & -1 & 0 \\
-1 & -3 & 3 & -1 \\
2 & -5 & 2 & -3 \\
-1 & 1 & 0 & 1
\end{pmatrix}
\]

#### 2. Next, add multiples of the first row to the others to eliminate the first column elements below the pivot.

- \(R_2 \rightarrow R_2 + R_1\)
- \(R_3 \rightarrow R_3 - 2R_1\)
- \(R_4 \rightarrow R_4 + R_1\)

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

#### 3. Now, proceed with the second pivot (second element in the second row). 

Divide Row 2 by \(-2\) to make the pivot element \(1\).

\[
R_2 \rightarrow -\frac{1}{2} R_2:
\begin{pmatrix}
1 & 1 & -1 & 0 \\
0 & 1 & -1 & \frac{1}{2} \\
0 & -7 & 4 & -3 \\
0 & 2 & -1 & 1
\end{pmatrix}
\]

#### 4. Eliminate the elements below and above the second pivot (second column).

- \(R_3 \rightarrow R_3 + 7R_2\)
- \(R_4 \rightarrow R_4 - 2R_2\)
- \(R_1 \rightarrow R_1 - R_2\)

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

#### 5. Make the third pivot \(1\) by dividing Row 3 by \(-3\).

\[
R_3 \rightarrow -\frac{1}{3} R_3:
\begin{pmatrix}
1 & 0 & 0 & -\frac{1}{2} \\
0 & 1 & -1 & \frac{1}{2} \\
0 & 0 & 1 & -\frac{1}{6} \\
0 & 0 & 1 & 0
\end{pmatrix}
\]

#### 6. Finally, eliminate the element above the third pivot in Row 2.

\[
R_2 \rightarrow R_2 + R_3:
\begin{pmatrix}
1 & 0 & 0 & -\frac{1}{2} \\
0 & 1 & 0 & \frac{1}{3} \\
0 & 0 & 1 & -\frac{1}{6} \\
0 & 0 & 0 & \frac{1}{6}
\end{pmatrix}
\]

### Step 2: Identify Echelon Form and Rank

The matrix is now in echelon form. The rank of the matrix is the number of non-zero rows.

#### Rank of Matrix: **4** (since all rows are non-zero).

Would you like more details on any of the steps? Here are some related questions you might find interesting:

1. What are the types of row operations in Gaussian elimination?
2. How do you interpret the rank of a matrix in terms of linear independence?
3. What is the significance of a matrix's rank in determining the solutions of a linear system?
4. How can you determine if a matrix is singular using its echelon form?
5. How does Gaussian elimination help in finding the inverse of a matrix?

**Tip:** The rank of a matrix is crucial in determining whether a system of linear equations has a unique solution, infinitely many solutions, or no solution at all.

Learn how to reduce a matrix to echelon form and find its rank with detailed step-by-step instructions. This problem covers matrix operations, Gaussian elimination, and understanding the rank of a matrix. Suitable for advanced high school students or anyone interested in linear algebra concepts.

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