Question

find the value of Determinant:
| 1 4 9 16 |
| 4 9 16 25 |
| 9 16 25 36 |
|16 25 36 49 |

Answer 1

Wow, a 4x4 with big numbers!

Answer 2

The basic method is to find the determinants of four 3x3 matrices and add them up.
Here the first term is
1 x the determinant of
9 16 25
16 25 36
25 36 49

second term is 4 times the determinant
4 16 25
9 25 36
16 36 49

the sign of the second and fourth terms will be negative according to the checkerboard rule.
http://en.wikipedia.org/wiki/Determinant

perhaps there is a way to make things cancel, but I don't know it.

Answer 3

Note that this determinant can be written
|1^2 2^2 3^2 4^2 |
|2^2 3^2 4^2 5^2 |
|3^2 4^2 5^2 6^2 |
|4^2 5^2 6^2 7^2 |
Subtract the first from the second row, second from the third, ... and the fourth row of the first.
Use well know formula a^2-b^2 = (a-b)(a+b)
After this, the determinant takes the form
|(4-1)(4+1) (5-2)(5+2) (6-3)(6+3) (7-4)(7+4) |
|(2-1)(2+1) (3-2)(3+2) (4-3)(4+4) (5-4)(5+4) |
|(3-2)(3+2) (4-3)(4+3) (5-4)(5+4) (6-5)(6+5) |
|(4-3)(4+3) (5-4)(5+4) (6-5)(6+5) (7-6)(7+6) |
After simplify:
|3*5 3*7 3*9 3*11 |
| 3 5 7 9 | =0
| 5 7 9 11 |
| 7 9 11 13 |

Answer 4

Two line are equal

Answer 5

@athe That's just a wonder piece of mathematics. I stand awestruck.

Answer 6

@zayesha If you are working a problem like this I suppose you will know this, but perhaps not. What @athe did was to show that he could manipulate to where one row is a multiple of another. In that situation the determinant must be equal to zero.

An easy way to see why is to just do the subtraction that zeroes out one of the rows, then since the "minors" can be chosen based on any row, we use the zeroed row, and all terms are clearly equal to 0.

Answer 7

Thanks for explanation @telliott99! Laziness was spread on the row :) and The determinant was already very symmetrical! ;)