Question

You have a Square table and it is filled with natural numbers in sequence. Two of them are given for your reference 3X3 1 2 3 4 5 6 7 8 9 4X4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 In the above quad sum of number at diagonal is 6+11 = 17 Now using above table do the following - 1. Select any quad of numbers (2X2)( as shown in pic ) and find out if there is any pattern you see in sum of diagonal elements 2. To get the answer you can make more such square table like 5X5 and 6X6 type. 3. Can you formulate an algebraic equation for above pattern for NXN type of a square table where N represent no of rows & columns? 4. Is there any pattern visible if you take difference between sum of diagonal element in a chosen quad? 5. Can you write an algebraic equation for it too

Answer 1

Steps used:

1) We will calculate no. of 2x2 quad in case of 3x3, 4x4,5x5,..

matrix .

Results : No. of quads of 2x2 :

In 3x3 matrix = 4

In 4x4 matrix = 9

In 5x5 matrix =16 .

Now, for N x N , we can say No. of 2x2 quads will be

(N-1)*(N-1) .

Similarly, we can find series in case of trace of matrices .

Then , we will generalize it for N x N matrix . F

For more details, refer to pics attached .  

For No. of traces in case of N x N will be caculated by using concept of progression and Sum of series .