If $A = \left[ {\begin{array}{*{20}{c}}
  1&2&3 \\
  5&0&7 \\
  6&2&5
\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}
  1&3&5 \\
  0&0&2
\end{array}} \right]$ , then which of the following is defined?
A. $AB$
B. $A + B$
C. $A'B'$
D. $B'A'$

Hint: Matrix Addition is defined only for matrices with the same order and Matrix Multiplication, for two matrices, is defined when the number of columns of the first matrix is equal to the number of rows of the second matrix.

Complete step by step Solution:
Given Matrices:
 $A = \left[ {\begin{array}{*{20}{c}}
  1&2&3 \\
  5&0&7 \\
  6&2&5
\end{array}} \right]$
And
$B = \left[ {\begin{array}{*{20}{c}}
  1&3&5 \\
  0&0&2
\end{array}} \right]$
Let us first determine the order of the two matrices.
Order of Matrix $A = 3 \times 3$
Order of Matrix $B = 2 \times 3$
Clearly, $A + B$ is not defined as Matrix Addition is only defined for matrices with the same order.
Also, $AB$ is not defined for Matrix Multiplication to be defined, the number of columns of $A$ should be equal to the number of rows of $B$.
Let us now determine the order of the transpose of the two matrices.
Order of Matrix $A' = 3 \times 3$
Order of Matrix $B' = 3 \times 2$
Clearly, $A'B'$ is defined while $B'A'$ is not defined.
Hence, only $A'B'$ is defined for the given matrices $A$ and $B$.

Hence, the correct option is (C).

Note:Transpose of a matrix is calculated by interchanging either the rows with the columns or the columns with the rows. It is denoted by $X'$. Thus, if the order of a matrix $X$ is $\left( {m \times n} \right)$ , then the order of matrix $X'$ is \[\left( {n \times m} \right)\].