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Let's go through the problems in this image, beginning with the first one.

### Problem 1: Rank of the Matrix

We are given the matrix:
\[
A = \begin{pmatrix}
-2 & -1 & -3 & -1 \\
1 & 2 & 3 & 1 \\
0 & 1 & 0 & 1
\end{pmatrix}
\]
We need to reduce this matrix to its row echelon form to find the rank. Let me work on this reduction.

### Step 1: Original Matrix
\[
A = \begin{pmatrix}
-2 & -1 & -3 & -1 \\
1 & 2 & 3 & 1 \\
0 & 1 & 0 & 1
\end{pmatrix}
\]

### Step 2: Row Operations to Reduce to Row Echelon Form
We will perform elementary row operations to transform the matrix into upper triangular form.

1. Multiply Row 1 by \( -\frac{1}{2} \):
   \[
   R_1 \to \frac{1}{2}R_1: \begin{pmatrix} 1 & \frac{1}{2} & \frac{3}{2} & \frac{1}{2} \end{pmatrix}
   \]

2. Subtract Row 1 from Row 2 to eliminate the first element in the second row:
   \[
   R_2 \to R_2 - R_1: \begin{pmatrix} 0 & \frac{3}{2} & \frac{3}{2} & \frac{1}{2} \end{pmatrix}
   \]

3. The third row already has a 0 in the first column, so no further operations are needed for that.

### Step 3: Continue Reduction
Next, we'll make the second row's leading coefficient 1 by multiplying by \( \frac{2}{3} \):
\[
R_2 \to \frac{2}{3}R_2: \begin{pmatrix} 0 & 1 & 1 & \frac{1}{3} \end{pmatrix}
\]

Finally, subtract \( \frac{1}{2} \) of Row 2 from Row 1 to eliminate the second column of Row 1:
\[
R_1 \to R_1 - \frac{1}{2}R_2: \begin{pmatrix} 1 & 0 & 1 & \frac{1}{3} \end{pmatrix}
\]

Thus, the row-echelon form is:
\[
\text{Row-echelon form of } A = \begin{pmatrix} 1 & 0 & 1 & \frac{1}{3} \\ 0 & 1 & 1 & \frac{1}{3} \\ 0 & 0 & 0 & 0 \end{pmatrix}
\]

### Step 4: Determine Rank
The rank is the number of non-zero rows in this row-echelon form. Since there are two non-zero rows, the **rank of matrix A is 2**.

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Would you like details for the next problems, or do you have any questions on this solution?

Here are 5 questions to expand on this topic:
1. What does the rank of a matrix represent in terms of its columns?
2. Can a matrix have a rank higher than its number of rows or columns?
3. How is the rank of a matrix related to the solution of a system of linear equations?
4. What is the difference between row echelon form and reduced row echelon form?
5. How does the rank relate to the determinant of a matrix?

**Tip**: The rank of a matrix gives you insight into the number of linearly independent rows or columns, which is key for determining the solution space of associated linear systems.

Find the rank of the matrix A = [ [-2, -1, -3, -1], [1, 2, 3, 1], [0, 1, 0, 1] ] by reducing it to row echelon form.

Learn how to find the rank of the matrix A = [ [-2, -1, -3, -1], [1, 2, 3, 1], [0, 1, 0, 1] ] by reducing it to row echelon form using Gaussian elimination. This problem covers linear algebra concepts such as matrix rank and the Rank-Nullity Theorem.

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