The rank of the matrix `A={:[(1,2,3),(3,6,9),(1,2,3)]:}`, is

Options

A. 1
B. 2
C. 3
D. none of these

Explanation

To find the rank of the matrix \( A = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 6 & 9 \\ 1 & 2 & 3 \end{pmatrix} \), we will use the method of row reduction to echelon form. The rank of a matrix is defined as the number of non-zero rows in its row echelon form.

### Step-by-step Solution:

1. **Write the Matrix:**
\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 6 & 9 \\ 1 & 2 & 3 \end{pmatrix} \]

2. **Perform Row Operations:**
- We will start by eliminating the first element of the second and third rows using the first row.
- For \( R_2 \), we can replace it with \( R_2 - 3R_1 \):
\[ R_2 = R_2 - 3R_1 = \begin{pmatrix} 3 & 6 & 9 \end{pmatrix} - 3 \begin{pmatrix} 1 & 2 & 3 \end{pmatrix} = \begin{pmatrix} 3 - 3 & 6 - 6 & 9 - 9 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \end{pmatrix} \]
- For \( R_3 \), we can replace it with \( R_3 - R_1 \):
\[ R_3 = R_3 - R_1 = \begin{pmatrix} 1 & 2 & 3 \end{pmatrix} - \begin{pmatrix} 1 & 2 & 3 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \end{pmatrix} \]

3. **Update the Matrix:**
After performing the row operations, the matrix \( A \) becomes:
\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]

4. **Count Non-Zero Rows:**
- In the updated matrix, we can see that there is only **one non-zero row** (the first row).
- The second and third rows are all zeros.

5. **Determine the Rank:**
- The rank of the matrix is equal to the number of non-zero rows, which is **1**.

### Final Answer:
The rank of the matrix \( A \) is **1**.

---