If A is a non-zero column matrix of order `mxx1` and B is a non-zero row matrix order `1xxn`, then rank of AB equals

Options

A. m
B. n
C. 1
D. none of these

Answer

To find the rank of the product of a non-zero column matrix \( A \) of order \( m \times 1 \) and a non-zero row matrix \( B \) of order \( 1 \times n \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Matrices**:
- Let \( A \) be a non-zero column matrix:
\[ A = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_m \end{pmatrix} \]
- Let \( B \) be a non-zero row matrix:
\[ B = \begin{pmatrix} b_1 & b_2 & \ldots & b_n \end{pmatrix} \]

2. **Calculate the Product \( AB \)**:
- The product \( AB \) will be a matrix of order \( m \times n \):
\[ AB = A \cdot B = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_m \end{pmatrix} \cdot \begin{pmatrix} b_1 & b_2 & \ldots & b_n \end{pmatrix} \]
- This results in:
\[ AB = \begin{pmatrix} a_1 b_1 & a_1 b_2 & \ldots & a_1 b_n \\ a_2 b_1 & a_2 b_2 & \ldots & a_2 b_n \\ \vdots & \vdots & \ddots & \vdots \\ a_m b_1 & a_m b_2 & \ldots & a_m b_n \end{pmatrix} \]

3. **Determine the Rank of \( AB \)**:
- The rank of a matrix is defined as the maximum number of linearly independent row vectors or column vectors in the matrix.
- Since \( A \) is a non-zero column matrix, it contributes one linearly independent column to \( AB \).
- Similarly, since \( B \) is a non-zero row matrix, it contributes one linearly independent row to \( AB \).
- Therefore, the rank of the product \( AB \) is equal to 1, as both matrices contribute one linearly independent vector.

4. **Conclusion**:
- Thus, the rank of the matrix \( AB \) is:
\[ \text{rank}(AB) = 1 \]