The rank of the matrix `A={:[(1,2,3,4),(4,3,2,1)]:}`, is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To find the rank of the matrix \( A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{pmatrix} \), we will use elementary row transformations to reduce the matrix to its row echelon form. The rank of the matrix is equal to the number of non-zero rows in this form.

### Step-by-Step Solution:

1. **Write the Matrix**:
\[ A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{pmatrix} \]

2. **Perform Row Operation**:
We will perform the operation \( R_2 \leftarrow R_2 - 4R_1 \) to eliminate the first element of the second row.
\[ R_2 = 4 - 4 \cdot 1 = 0 \]
\[ R_2 = 3 - 4 \cdot 2 = 3 - 8 = -5 \]
\[ R_2 = 2 - 4 \cdot 3 = 2 - 12 = -10 \]
\[ R_2 = 1 - 4 \cdot 4 = 1 - 16 = -15 \]
Thus, the matrix becomes:
\[ A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 0 & -5 & -10 & -15 \end{pmatrix} \]

3. **Simplify the Second Row**:
We can simplify the second row by factoring out -5:
\[ R_2 \leftarrow -\frac{1}{5} R_2 \]
This gives us:
\[ A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \end{pmatrix} \]

4. **Determine the Rank**:
Now, we have the matrix in row echelon form:
\[ A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \end{pmatrix} \]
We can see that there are 2 non-zero rows.

5. **Conclusion**:
Therefore, the rank of the matrix \( A \) is:
\[ \text{Rank}(A) = 2 \]