Some solved examples of applied mathematics are mentioned below:
Example 1: Determine if the following transition matrix is ergodic Markov chain.
Solution: Here we must check that it is possible to go from every present state to all other states. We observe that from state 1, it is possible to go directly to every other state except state 3. For state 3, it is possible to go from state 1 to state 2 to state 3. Therefore, it is possible to go from state 1 to any other state.
Similarly, from state 2, it is possible to go to state 3 or state 4, then from state 3 to state 1, or from state 4 to state 3 to state 1.
Also from state 3 it is possible to go directly to state 1. Finally from state 4, it is possible to go to state 3, then from state 3 to state 1.
Hence, the above transition matrix is an ergodic Markov chain.
Example 2: Find the range of the values of p and q which will render the entry (2, 2) a saddle point for the game:
Solution: First ignoring the values of p and q we will determine the maximin and the minimax values of the payoff matrix as below:
Since the entry (2, 2) is a saddle point, thus maximin value v = 7, minimax value v = 7. This imposes the condition on p as p $\leq$ 7 and on q as q $\geq$ 7. Hence, the range of p and q will be p $\leq$ 7 and q $\geq$ 7.
Example 3: Solve the following minimization problem:
Minimize p = 3x + y
subject to
2x  y $\leq$ 4
2x + 3y $\leq$ 12
y $\leq$ 3
x $\geq$ 0
y $\geq$ 0
Solution: After solving the lines, taking the vertex and the lines through the vertex, along with the value of their objective function, we will get the following table:
Vertex

Lines Through Vertex 
Value of Objective 
(3, 2) 
2x  y = 4; 2x + 3y = 12 
11 
(2, 0) 
2x  y = 4; y = 0 
6 
(1.5, 3) 
2x + 3y = 12; y = 3 
7.5 
(0, 3) 
y = 3; x = 0 
3 Minimum 
(0, 0) 
x = 0; y = 0 
0 Minimum 
Thus, we got the minimum value of the function at x = 0 and y = 3, hence, the minimum value is
3). However, if we can take the value (0, 0), then the minimum value of any objective function will always be zero.
The graph of the lines drawn can be shown as below:Here, note that the region shown in the white part is the feasible region only, not the red shaded part.