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 follow these steps:

### Step 1: Write the matrix
We start with the given matrix:
\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 6 & 9 \\ 1 & 2 & 3 \end{pmatrix} \]

### Step 2: Identify linearly dependent rows
Observe the rows of the matrix. The second row \( (3, 6, 9) \) is a multiple of the first row \( (1, 2, 3) \) since \( (3, 6, 9) = 3 \times (1, 2, 3) \). Additionally, the third row is identical to the first row. Therefore, we can conclude that the rows are linearly dependent.

### Step 3: Reduce the matrix to row echelon form
To find the rank, we can perform row operations to simplify the matrix. We can subtract 3 times the first row from the second row:
\[ R_2 = R_2 - 3R_1 \implies A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 1 & 2 & 3 \end{pmatrix} \]
Next, we can subtract the first row from the third row:
\[ R_3 = R_3 - R_1 \implies A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]

### Step 4: Count the non-zero rows
In the row echelon form, we see that there is only one non-zero row. Therefore, the rank of the matrix \( A \) is 1.

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

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