If \[A = \left[ {\begin{array}{*{20}{c}}
  1&2&3
\end{array}} \right],B = \left[ {\begin{array}{*{20}{c}}
  2 \\
  3 \\
  4
\end{array}} \right]\,and\,C = \left[ {\begin{array}{*{20}{c}}
  1&5 \\
  0&2
\end{array}} \right]\], then which of the following is defined?
A.\[AB\]
B.\[BA\]
C.\[\left( {AB} \right).C\]
D.\[\left( {AC} \right).B\]

Hint: Multiplication of two matrices A and B is defined when the number of columns of A is equal to the number of rows of B. To multiply entries in rows by equivalent entries in columns, multiply the first row's entries by the first column's entries, and so on for all other entries.

Formula Used: If \[A = {[{a_{ij}}]_{m \times n}}\] and \[B = {[{b_{ij}}]_{n \times p}}\] then we can say that \[A \times B = C\] where the value of C is
\[C = {[{c_{ij}}]_{m \times p}}\]
Here \[{c_{ij}} = \mathop \sum \limits_{j = 1}^m {a_{ij}}{b_{jk}} = {a_{i1}}{b_{1k}} + {a_{i2}}{b_{2k}} + ........ + {a_{im}}{b_{mk}}\]

Complete step by step Solution:
We are given that,
\[A = \left[ {\begin{array}{*{20}{c}}
  1&2&3
\end{array}} \right],B = \left[ {\begin{array}{*{20}{c}}
  2 \\
  3 \\
  4
\end{array}} \right]\,and\,C = \left[ {\begin{array}{*{20}{c}}
  1&5 \\
  0&2
\end{array}} \right]\]
Since in A and B, the number of columns of A is equal to the number of rows of B. Multiplication of A and B is defined. Therefore, we have,
\[AB = {\left[ {\begin{array}{*{20}{c}}
  1&2&3
\end{array}} \right]_{1 \times 3}}{\left[ {\begin{array}{*{20}{c}}
  2 \\
  3 \\
  4
\end{array}} \right]_{3 \times 1}}\]= \[{\left[ {20} \right]_{1 \times 1}}\]
Similarly in B and A, the number of columns in B is equal to the number of rows in A. The multiplication of B and A is defined. Therefore, we have,
\[BA = {\left[ {\begin{array}{*{20}{c}}
  2 \\
  3 \\
  4
\end{array}} \right]_{3 \times 1}}{\left[ {\begin{array}{*{20}{c}}
  1&2&3
\end{array}} \right]_{1 \times 3}} = {\left[ {\begin{array}{*{20}{c}}
  2&4&6 \\
  3&6&9 \\
  4&8&{12}
\end{array}} \right]_{3 \times 3}}\]
In AB and C, the number of columns of AB is 1 which is not equal to the number of rows of C which is 2. Therefore, the Multiplication of AB and C is not defined.
In A and C, the number of columns of A which is 3 is not equal to the number of rows of C which is 2. Hence Multiplication of A and C is not defined. Therefore, the multiplication of AC and B is not defined.

Therefore, the correct option is (A) and (B).

Note:It is important to note that matrix multiplication is defined in a complex manner. It is not just multiplying each respective element of both matrices like addition. In Matrix Multiplication, we multiply an m × n matrix with n × p matrix to get m × p matrix.