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 matrix product \( AB \) where \( A \) is a non-zero column matrix of order \( m \times 1 \) and \( B \) is a non-zero row matrix of order \( 1 \times n \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the matrices**:
- Let \( A \) be a column matrix represented as:
\[ A = \begin{pmatrix} A_1 \\ A_2 \\ \vdots \\ A_m \end{pmatrix} \]
where \( A_i \) are the elements of the matrix \( A \).
- Let \( B \) be a row matrix represented as:
\[ B = \begin{pmatrix} B_1 & B_2 & \cdots & B_n \end{pmatrix} \]
where \( B_j \) are the elements of the matrix \( B \).

2. **Multiply the matrices**:
- The product \( AB \) will be a matrix of order \( m \times n \) and can be calculated as follows:
\[ AB = \begin{pmatrix} A_1 \cdot B_1 & A_1 \cdot B_2 & \cdots & A_1 \cdot B_n \\ A_2 \cdot B_1 & A_2 \cdot B_2 & \cdots & A_2 \cdot B_n \\ \vdots & \vdots & \ddots & \vdots \\ A_m \cdot B_1 & A_m \cdot B_2 & \cdots & A_m \cdot B_n \end{pmatrix} \]

3. **Analyze the resulting matrix**:
- Since both \( A \) and \( B \) are non-zero matrices, at least one element in \( A \) (say \( A_i \)) and at least one element in \( B \) (say \( B_j \)) are non-zero.
- Therefore, the product \( A_i \cdot B_j \) will also be non-zero, which means that \( AB \) will have at least one non-zero entry.

4. **Determine the rank**:
- The rank of a matrix is defined as the maximum number of linearly independent row or column vectors in the matrix.
- Since \( AB \) has at least one non-zero entry, it implies that the rank of \( AB \) is at least 1.
- However, since \( A \) is a column matrix and \( B \) is a row matrix, the resulting matrix \( AB \) can be expressed as a linear combination of the columns of \( A \) scaled by the entries of \( B \). This means that all rows of \( AB \) are linearly dependent on each other.
- Therefore, the rank of \( AB \) is exactly 1.

### Conclusion:
The rank of the matrix product \( AB \) is:
\[ \text{rank}(AB) = 1 \]