# Determinant of a Finite Field 2x2 Matrix

## 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
Science Advisor
Homework Helper

## 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.

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
Science Advisor
Homework Helper
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.

Ok then,

Thanks!