Rank of a Matrix:
The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. In other words, it represents the maximum number of rows or columns that can be selected without any redundancy or duplication.

Multiplication of Matrices:
When two matrices are multiplied, the resulting matrix will have dimensions equal to the number of rows in the first matrix and the number of columns in the second matrix. In the given scenario, we are multiplying a 3 x 1 column matrix A with a 1 x 3 row matrix B, resulting in a 3 x 3 matrix C.

Rank of C (AB):
To find the rank of matrix C, we need to determine the number of linearly independent rows or columns in C. Since C is a 3 x 3 matrix, we need to find the maximum number of linearly independent rows or columns that can be selected from C.

Matrix A:
Since matrix A is a 3 x 1 column matrix, it has 3 rows and 1 column. The maximum number of linearly independent rows or columns that can be selected from A is 1, as it has only 1 non-zero column.

Matrix B:
Similarly, matrix B is a 1 x 3 row matrix, meaning it has 1 row and 3 columns. The maximum number of linearly independent rows or columns that can be selected from B is also 1, as it has only 1 non-zero row.

Multiplication of A and B:
When matrix A (3 x 1) is multiplied by matrix B (1 x 3), the resulting matrix C (3 x 3) will have elements obtained by multiplying corresponding elements of A and B.

Since both matrix A and matrix B have only 1 non-zero row/column, the resulting matrix C will also have only 1 linearly independent row/column.

Conclusion:
Therefore, the rank of the matrix C (AB) is 1, as it can have a maximum of 1 linearly independent row or column. Hence, the correct answer is option 'B'.