Question

P1. 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
Now using above table do the following
1. Select many vertical pairs ( 2 numbers vertically ) and find difference between them try and observe any similarity between them.
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 NXN type of a square table where N represent no of rows & columns?
4. Repeat same exercise for difference of numbers in a horizontal pair
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 .