Question about determinants (restated)

Look at this:
1.
The two numbers 21 and 12 are divisible by 3. The determinant

| 2 1 |
| 1 2 |

is equal to 2*2-1*1=3, which is divisible by 3.

Or for another example, the three numbers 112, 763, and 959 are divislbe by 7.

The determinant

|1 1 2|
|7 6 3|
|9 5 9|

= 6*9 + 3*9 + 2*7*5 - 9*6*2 - 5*3 - 7*9 = 54 + 27 + 70 - 108 - 15 - 63 = -35 which is also divisible by 7.

The question is this: does this always work?

|3 3|
|0 0|

Kaboom...

Having said that, I tried a bunch of 2x2 matrices with no zeroes, and making sure that det > 0, and it worked (for divisible by 3 case). You might be onto something.

It doesn't really have to be determinants though, as I don't think any special property of matrices is causing it. Reformulate it as an equation (writing, for example, 12 as 1*10 + 2*1, and so on) and mess about with it. It'll just be a slightly rearranged version of result that if the digit sum of a number is divisible by three, then the number is divisible by three.