I m solving the first question .See here in this question nothing specific about matrix needs to be kept into mind.Just a pattern needs to be recognised.
See the ratio : q31 + q32 / q21 is being asked .It is interesting to note that we perform multiplication on the given matrix itself and find that only these 3 elements value are being updated the rest of the values (upper triangular matrix) values will remain same inclusive of diagonal elements so we need not bother about them.
When we multiply once the matrix P given with itself ,
we get for P2 matrix : q21 = 8 = 4 * 2 [4 was the value of q21 in P]
q31 = 48 = 8 * 6 [whereas in P we had 16 in this place which can be written as 16 = 4 * 4]
q32 = 8 = q31
Similarly on computing for P3 , we get :
q21 = 12 = 4 * 3
q31 = 96 = 12 * 8
q32 = 12 = q31
So we can now generalise for P50 , i.e. we need to find the 50 th term of each of the above 3 as individually they are forming a series
We can write : q21 = q32 = 4 * 50 = 200
q31 = 200 * [4 + 49 * 2]
= 200 * 102
= 20400
So now given P50 - I = Q ( I have transposed Q to RHS and I to LHS from question] where I is the identity matrix .Still there is no effect on concerned term because those terms of identity matrix are non diagonal and hence remain 0 so no effect on the above 3 elements found hence.
Hence , the ratio : (q31 + q32) / q21
= (20400 + 200) / 200
= 103
Hence B) is the correct answer.
