Prime Field Homework: Find A-1 Over Z5

In summary, to find A-1 over the prime field Z5, we need to use the adjoint method to find the inverse of A and then reduce all of its values to (mod 5) form. This can be done by solving for the coefficients in the equation 16= 7x (mod 5) and substituting them into the inverse matrix. The final matrix will be [401, 321, 314].
  • #1
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Homework Statement



Let A = [3 2 1; 0 1 4; 4 2 1]

(i) Find the cofactors C11, C12, C13, C21, C22, C23, C31, C32, C33 of A.
(ii) Given that det(A) = 7, use the adjoint method to find A-1.
(iii) Use the answer to part (ii) to find A-1 over the prime field Z5.

The Attempt at a Solution



For (i) and (ii) I got inverse of A = 1/7[-7 0 7; 16 -1 -12; -4 2 3]
no problems here,

Just wondering how I do part (iii)
I've had a peek at the solutions and relevant textbooks, but can't seem to find anything.

Thanks
 
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  • #2
A "prime field" contains only positive integers. In particular, Z5 contains only {0, 1, 2, 3, 4}. That is, it does not explicitely contain "16/7" or -1/7. For that matter, it does not contain "-7/7= -1". But since 1+ 6= 7= 0 (mod 5), "-1" is really just "6". Knowing that 16/7= x (mod 5) is the same as 16= 7x what is x? (Remember that 16= 3(5)+ 1 so 16= 1 (mod 5) and 7= 1(5)+ 2 so 7= 2 (mod 5). 16= 7x (mod 5) is the same as 1= 2x (mod 5). The only possible values of x are 0, 1, 2, 3, 4 so just try each.

Reduce all of the values of A to (mod 5) form.
 
  • #3
I understand the concept of mod 5, but I cannot comprehend all of it.

I can only make sense up to this part:

For that matter, it does not contain "-7/7= -1". But since 1+ 6= 7= 0 (mod 5)

How does 7=0?
isn't 7mod5=0?

"-1" is really just "6"
:confused:

1= 2x (mod 5). The only possible values of x are 0, 1, 2, 3, 4 so just try each

Am I trying to satify the equation using only x values ranging from 1 to 4? Whole numbers only?
 
  • #4
ok consider the first entry of column 1

1/7 (-7)
implies
-7 = 7a then mod 5 this

so
3 = 2a
note "a" can equal 0 1 2 3 4
so
3 = 2 * 0 = 0 mod 5 = 0 not true
3 = 2 * 1 = 2 mod 5 = 2 not true
3 = 2 * 2 = 4 mod 5 = 4 not true
3 = 2 * 3 = 6 mod 5 = 1 not true
3 = 2 * 4 = 8 mod 5 = 3 true
so first entry is 4

consider the second entry of column 1

16 / 7
implies
-7 = 7a then mod 5 this

so
1 = 2a
note again "a" can equal 0 1 2 3 4
so
1 = 2 * 3 = 6 mod 5 = 1 true

the resultant matrix should be

[401,321,314]
 

1. What is a prime field?

A prime field is a mathematical structure that consists of a set of numbers, also known as elements, and two operations (addition and multiplication) that satisfy certain properties. In a prime field, the numbers are limited to prime numbers and the operations follow specific rules.

2. What does "A-1 Over Z5" mean?

"A-1 Over Z5" refers to finding the inverse of a number, denoted as A, in the prime field Z5. This means finding a number that, when multiplied by A, results in the multiplicative identity element in Z5, which is 1.

3. Why is finding the inverse in a prime field important?

Finding the inverse in a prime field is important because it allows for division, which is a fundamental operation in mathematics. It also allows for solving equations and finding solutions to problems in various fields, such as cryptography and coding theory.

4. How do you find the inverse of a number in Z5?

To find the inverse of a number A in Z5, you can use the Extended Euclidean Algorithm or the Fermat's Little Theorem. With the Extended Euclidean Algorithm, you can find the inverse using a step-by-step process, while Fermat's Little Theorem provides a formula for finding the inverse directly.

5. What is an example of finding the inverse over Z5?

An example of finding the inverse over Z5 is finding the inverse of 3. Using the Extended Euclidean Algorithm, the process would be:3 = 1*3 + 0*55 = 0*3 + 1*51 = 5 - 0*31 = 51*3 = 5*3 - 0*31*3 = 5*3Therefore, the inverse of 3 in Z5 is 2, since 2*3 = 1 (mod 5).

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