If $\left[ \begin{matrix}
   m & n \\
\end{matrix} \right]$$\left[ \begin{matrix}
   m \\
   n \\
\end{matrix} \right]$= $\left[ 25 \right]$ and m < n, then $\left[ m,n \right]$ =
A . $\left[ 2,3 \right]$
B . $\left[ 3,4 \right]$
C . $\left[ 4,3 \right]$
D . None of these

Hint: In this question, we have given matrices in the form of m and n and we have to find the value of m and n. First we multiply both the matrix and then by putting the values of m and n which are given in the options, check whether which value of m and n satisfies the given equation and choose the correct option.
Complete Step- by- step Solution:
Given $\left[ \begin{matrix}
   m & n \\
\end{matrix} \right]$$\left[ \begin{matrix}
   m \\
   n \\
\end{matrix} \right]$= $\left[ 25 \right]$
First we multiply the matrices, we get
${{m}^{2}}+\,{{n}^{2}}=25$…………………………………… [1]
Now we use the options to find the value of m and n.
We put the values of m and n is equation [1] and check whether which value satisfies the equation.
First let $m=2,n=3$
Then ${{[2]}^{2}}+\,{{[3]}^{2}}=25$
Solving further, we get
$13\ne 25$
Then we put $m=3,n=4$
Then ${{[3]}^{2}}+\,{{[4]}^{2}}=25$
Hence $25=25$
In option [ C ] where $m=4,n=3$
But in this question, we have given m < n
So $m=4,n=3$ value does not possible.
After checking put all the options, we found that the value of $\left[ m,n \right]$ = $\left[ 3,4 \right]$

Thus, Option [ B ] is the correct answer.

Note: In these types of questions, Students male mistakes in multiplying the matrices. In multiplication of matrices, if we take $A[4\times 3]$ and $B[3\times 4]$ multiplication of both the matrix is possible but If we take $A[4\times 3]$ and $B[4\times 3]$ multiplication of matrices is not possible.