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Find the rank of the matrix using echelon form.

To determine the rank of the given matrix \( A \), we first need to reduce it to its row echelon form (REF). The matrix \( A \) is:

\[
A = \begin{pmatrix}
3 & 4 & 3 & 1 \\
2 & 0 & 2 & 3 \\
0 & 6 & 2 & 2 \\
1 & 2 & 2 & 8
\end{pmatrix}
\]

The steps for row reduction will involve using elementary row operations to simplify the matrix. Let's begin.

### Step 1: Make the first element in the top-left position (pivot) equal to 1, or work with it as the pivot.

1. We can leave row 1 as it is for now:
\[
R_1 = \begin{pmatrix} 3 & 4 & 3 & 1 \end{pmatrix}
\]
2. Subtract appropriate multiples of row 1 from the other rows to create zeros below the first pivot.

Perform row operations:
\[
R_2 = R_2 - \frac{2}{3}R_1
\]
\[
R_3 = R_3 - 0R_1
\]
\[
R_4 = R_4 - \frac{1}{3}R_1
\]

Let's perform these operations and continue reducing to echelon form.After the first set of row operations, the matrix becomes:

\[
\begin{pmatrix}
3 & 4 & 3 & 1 \\
0 & -2.67 & 0 & 2.33 \\
0 & 6 & 2 & 2 \\
0 & 0.67 & 1 & 7.67
\end{pmatrix}
\]

### Step 2: Focus on the second pivot (second row, second column).

We now work on the second row and aim to create zeros below the second pivot. The current pivot is \(-2.67\), so we'll use this pivot for the next row operations.

Perform row operations:
\[
R_3 = R_3 - \frac{6}{-2.67}R_2
\]
\[
R_4 = R_4 - \frac{0.67}{-2.67}R_2
\]

Let's proceed with these operations.After the second set of row operations, the matrix becomes:

\[
\begin{pmatrix}
3 & 4 & 3 & 1 \\
0 & -2.67 & 0 & 2.33 \\
0 & 0 & 2 & 7.25 \\
0 & -0.0033 & 1 & 8.25
\end{pmatrix}
\]

### Step 3: Focus on the third pivot (third row, third column).

Now we aim to create zeros below the third pivot. The current pivot is \(2\). We will subtract appropriate multiples of the third row from the fourth row to eliminate the \(1\) in the fourth row.

Perform the row operation:
\[
R_4 = R_4 - \frac{1}{2}R_3
\]

Let's execute this next operation.The matrix in row echelon form is now:

\[
\begin{pmatrix}
3 & 4 & 3 & 1 \\
0 & -2.67 & 0 & 2.33 \\
0 & 0 & 2 & 7.25 \\
0 & -0.0033 & 0 & 4.63
\end{pmatrix}
\]

### Step 4: Determine the rank.

In this form, we count the number of non-zero rows (rows with at least one non-zero element). All four rows contain non-zero elements, which means that the **rank of the matrix is 4**.

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

### Further questions:
1. What are the steps to find the inverse of this matrix, if possible?
2. How would you solve a system of linear equations using this matrix?
3. Can this matrix be reduced to reduced row echelon form (RREF)?
4. How do the eigenvalues of this matrix relate to its rank?
5. What is the determinant of this matrix?

### Tip:
When performing row operations, always focus on clearing entries below the pivot elements to move systematically toward echelon form.

Learn how to find the rank of the given 4x4 matrix by reducing it to row echelon form. Step-by-step solution involves performing elementary row operations. Suitable for undergraduate students studying linear algebra.

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