A study of students taking a 20-question exam ranked their progress from one testing period to the next. Students scoring 0 to 5 from group 1, those scoring 6 to 10 for group 2, those scoring 11 to 15 from group 3 and those scoring 15 to 20 form group 4. The transition matrix to the right shows the result.

**Solution:-**

Find the long-range prediction for the proportion of the students in each group.

Let V = [w x y z ] be the vector for the long-range predication. If P is the given matrix, solve the equation VP=V together with the equation w + x + y + z=1 to get w ,x ,y, and z.

w = 0.1247

x = 0.2325

y = 0.2464

z = 0.3964

The proportion of students will be 12.47% in group 1, 23.25% in group 2, 24.64 in group 3, and 39.64% in group 4.

How do you solve the problem?