Rank of a Matrix:
The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It represents the dimension of the vector space spanned by the rows or columns of the matrix.

Multiplication of Matrices:
When multiplying two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. 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.

Multiplication of a Column Matrix and a Row Matrix:
When multiplying a column matrix (size m x 1) with a row matrix (size 1 x n), the resulting matrix will have dimensions m x n.

Given Information:
In this question, we are given that matrix C is obtained by multiplying a non-zero column matrix A of size 3 x 1 and a non-zero row matrix B of size 1 x 3.

Size of Matrices:
The column matrix A has size 3 x 1, which means it has 3 rows and 1 column.
The row matrix B has size 1 x 3, which means it has 1 row and 3 columns.
The resulting matrix C will have size 3 x 3, which means it will have 3 rows and 3 columns.

Rank of Matrix C:
To determine the rank of matrix C, we need to find the maximum number of linearly independent rows or columns in matrix C.

Rank of a 3 x 3 Matrix:
For a 3 x 3 matrix, the maximum possible rank is 3. This means that the rank of matrix C can be at most 3.

Multiplication of A and B:
When multiplying matrix A (3 x 1) and matrix B (1 x 3), the resulting matrix C (3 x 3) will have the following form:

C = AB = (a1 * b1) + (a2 * b2) + (a3 * b3)

Here, a1, a2, and a3 are the elements of matrix A, and b1, b2, and b3 are the elements of matrix B.

Rank of C:
Since the matrices A and B are non-zero, at least one element in each matrix is non-zero. Therefore, when multiplying A and B, the resulting matrix C will have at least one non-zero element.

If there is at least one non-zero element in matrix C, then the rank of C will be at least 1.

Therefore, the correct answer is option 'B' - 1. The rank of the 3 x 3 matrix C, obtained by multiplying a non-zero column matrix A of size 3 x 1 and a non-zero row matrix B of size 1 x 3, is 1.