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

            1      2      3

A=       4      5      6

            7      8      9

            0      2      1
 

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

Given matrix \( A \):

\[
A = \begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 \\
0 & 2 & 1
\end{pmatrix}
\]

### Step 1: Subtract multiples of row 1 from rows 2 and 3 to create zeros below the pivot (the first element in row 1).
- Row 2: \( R_2 = R_2 - 4R_1 \)
- Row 3: \( R_3 = R_3 - 7R_1 \)

\[
\text{New } A = \begin{pmatrix}
1 & 2 & 3 \\
0 & -3 & -6 \\
0 & -6 & -12 \\
0 & 2 & 1
\end{pmatrix}
\]

### Step 2: Divide row 2 by -3 to normalize the pivot in the second row.
- Row 2: \( R_2 = \frac{R_2}{-3} \)

\[
\text{New } A = \begin{pmatrix}
1 & 2 & 3 \\
0 & 1 & 2 \\
0 & -6 & -12 \\
0 & 2 & 1
\end{pmatrix}
\]

### Step 3: Eliminate the non-zero entries in the third and fourth rows using row 2.
- Row 3: \( R_3 = R_3 + 6R_2 \)
- Row 4: \( R_4 = R_4 - 2R_2 \)

\[
\text{New } A = \begin{pmatrix}
1 & 2 & 3 \\
0 & 1 & 2 \\
0 & 0 & 0 \\
0 & 0 & -3
\end{pmatrix}
\]

### Step 4: Normalize the pivot in row 4.
- Row 4: \( R_4 = \frac{R_4}{-3} \)

\[
\text{New } A = \begin{pmatrix}
1 & 2 & 3 \\
0 & 1 & 2 \\
0 & 0 & 0 \\
0 & 0 & 1
\end{pmatrix}
\]

### Final Row Echelon Form:

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

### Conclusion:
The matrix has **3 non-zero rows** in its row echelon form, so the **rank of matrix \( A \) is 3**.

Do you want details on any of the steps, or have further questions?

Here are 5 related questions:
1. What is the significance of row echelon form in determining matrix rank?
2. How is Gaussian elimination related to solving systems of equations?
3. What is the difference between row echelon form and reduced row echelon form?
4. Can the rank of a matrix tell us about the invertibility of the matrix?
5. What happens if two rows of a matrix are proportional?

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

Determine the rank of the following matrices by reducing them to echelon form: A = [[1, 2, 3], [4, 5, 6], [7, 8, 9], [0, 2, 1]]

This detailed solution explains how to determine the rank of a matrix by reducing it to row echelon form using Gaussian elimination. The problem demonstrates key linear algebra concepts, including matrix rank, row operations, and the use of the Rank-Nullity Theorem. Suitable for Grades 11-12 and undergraduate students.

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