Find the rank of the following matrices:-

al [[1, 3, 7, 1, 2], [4, 0, 5, 2, 9], [3, 3, 4, 7, 4], [0, 0, 6, 6, - 3]]

To find the rank of the matrix, we perform Gaussian elimination to reduce it to row echelon form (REF). The rank of a matrix is the number of non-zero rows in its REF.

The given matrix is:
\[
\begin{bmatrix}
1 & 3 & 7 & 1 & 2 \\
4 & 0 & 5 & 2 & 9 \\
3 & 3 & 4 & 7 & 4 \\
0 & 0 & 6 & 6 & -3
\end{bmatrix}
\]

### Step 1: Perform Row Operations

We start with the matrix and use elementary row operations (subtracting multiples of rows) to create zeros below the pivot in each column.

1. \( R_2 = R_2 - 4R_1 \)
2. \( R_3 = R_3 - 3R_1 \)

This will update the matrix as follows:
\[
\begin{bmatrix}
1 & 3 & 7 & 1 & 2 \\
0 & -12 & -23 & -2 & 1 \\
0 & -6 & -17 & 4 & -2 \\
0 & 0 & 6 & 6 & -3
\end{bmatrix}
\]

### Step 2: Create Zero Below Next Pivot (Second Column)

We now focus on the second column. To make the zeros below the pivot (second row), perform the following:

3. \( R_3 = R_3 - \frac{1}{2} R_2 \)

This gives:
\[
\begin{bmatrix}
1 & 3 & 7 & 1 & 2 \\
0 & -12 & -23 & -2 & 1 \\
0 & 0 & -5.5 & 5 & -2.5 \\
0 & 0 & 6 & 6 & -3
\end{bmatrix}
\]

### Step 3: Work on Third Column

Finally, we make the matrix triangular by eliminating below the pivot in the third column:

4. \( R_4 = R_4 - \frac{6}{-5.5} R_3 \)

This simplifies to:
\[
\begin{bmatrix}
1 & 3 & 7 & 1 & 2 \\
0 & -12 & -23 & -2 & 1 \\
0 & 0 & -5.5 & 5 & -2.5 \\
0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]

### Step 4: Count Non-Zero Rows

In the final row echelon form, there are **3 non-zero rows**, meaning the rank of the matrix is:

\[
\text{Rank} = 3
\]

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Would you like further details on the row operations or a more in-depth explanation of the Gaussian elimination process?

### Related Questions:
1. How does the rank of a matrix relate to its solutions in linear systems?
2. Can you explain the difference between row echelon form (REF) and reduced row echelon form (RREF)?
3. What is the impact of having a zero row on the rank of a matrix?
4. How does Gaussian elimination help in solving linear systems?
5. Can the rank of a matrix be determined from its determinant?

### Tip:
The rank of a matrix is always less than or equal to the smallest dimension (rows or columns).

Find the rank of the following matrices: [[1, 3, 7, 1, 2], [4, 0, 5, 2, 9], [3, 3, 4, 7, 4], [0, 0, 6, 6, -3]]

This tutorial explains how to find the rank of a 4x5 matrix using Gaussian elimination. Follow the detailed step-by-step process of row operations and learn to reduce the matrix to its row echelon form (REF) to determine the rank. This problem is suitable for students in Grades 10-12 studying linear algebra.

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