Determinant of a Finite Field 2x2 Matrix

[rehcarlos]

Homework Statement

Find the determinant of:
|1 1|
|2 1|

The field is Z3.

Homework Equations

The field is Z3, that is, to multiply two numbers, you first multiply then take the remainder of the division by 3.

The Attempt at a Solution

I tried:
( 1 x 1 ) - ( 1 x 2 )
1 x 1 will be: 1x1 = 1 => then get the remainder of the division by 3 that is 1
1 x 2 will be: 1 x 2 = 2 => then get the remainder of the division by 3 that is 2

So, the determinant would be 1 - 2 = -1 => then get the remainder of the division by 3 that is -1

But the answer is 2

 

[Dick]

Homework Statement

Find the determinant of:
|1 1|
|2 1|

The field is Z3.

Homework Equations

The field is Z3, that is, to multiply two numbers, you first multiply then take the remainder of the division by 3.

The Attempt at a Solution

I tried:
( 1 x 1 ) - ( 1 x 2 )
1 x 1 will be: 1x1 = 1 => then get the remainder of the division by 3 that is 1
1 x 2 will be: 1 x 2 = 2 => then get the remainder of the division by 3 that is 2

So, the determinant would be 1 - 2 = -1 => then get the remainder of the division by 3 that is -1

But the answer is 2

In Z3, -1=2. Any two numbers that differ by a multiple of 3 have the same remainder when divided by 3.

 

[rehcarlos]

I know it may be simple, but I'm still confuse,

-1 divided by 3 has a remainder of -1
2 divided by 3 has a remainder of 2

So I don't get how they have the same remainder

 

[Dick]

I know it may be simple, but I'm still confuse,

-1 divided by 3 has a remainder of -1
2 divided by 3 has a remainder of 2

So I don't get how they have the same remainder

To say Z3 is about 'remainders' isn't quite accurate. Two number are equal in Z3 if they differ by a multiple of 3. That's the sense in which they have the same remainder. 2-(-1)=3*1.

 

[rehcarlos]

Ok then,

Thanks!