System of equations using modulo

• Firestrider
The inverse of the matrix \begin{bmatrix}6 & 17\\ 13 & 21\end{bmatrix} mod 26 is \begin{bmatrix}23 & 13\\ 15 & 6\end{bmatrix}.In summary, the inverse of the matrix \begin{bmatrix}6 & 17\\ 13 & 21\end{bmatrix} mod 26 is \begin{bmatrix}23 & 13\\ 15 & 6\end{bmatrix}.
Firestrider

Homework Statement

5) Determine the inverse of the matrix $$\begin{bmatrix} 6 & 17\\ 13 & 21 \end{bmatrix}$$ mod 26

Homework Equations

Solve for linear system of equations:
6a + 17c = 1 mod 26 6b + 17d = 0 mod 26
13a + 21c = 0 mod 26 13b + 21d = 1 mod 26

The Attempt at a Solution

6 * (13a + 21c) = 6 * (0 mod 26)
- 13 * (6a + 17c) = 13 * (1 mod 26)
--------------------------------------------
78a + 126c = 0 mod 26
- (78a + 221c = 13 mod 26)
--------------------------------------------
95c = 13 mod 26

This is where I'm stuck. How do you divide 13 mod 26 by 95?

Well, first you reduce 95 mod 26: 26 divides into 95 3 times with remainder 17 so 95= 17 (mod 26). Now you want to solve 17c= 13 (mod 26). You want to multiply both sides by the multiplicative inverse of 17 (mod 26). That, of course, would be an integer n such that 17n= 1 (mod 26). And that says that 17n must be one more than a multiple of 26: 17n= 26m+ 1 for some integer m.

Write that as the Diophantine equation 17n- 26m= 1. 17 divides into 26 once with remainder 9: 26- 17= 9. 9 divides into 17 once with remainder 8: 17- 9= 8. And 8 divides into 9 once with remainder 1: 9- 8= 1. Replace the "8" in that last equation with 17- 9: 9- (17- 9)= 2(9)- 17= 1. Replace the "9" in that with 26- 17: 2(26- 17)- 17= 2(26)- 3(17)= 1.

That is, n= -3 and m= -2 will work: 17(-3)- 26(-2)= 2(26)- 3(17)= 1. And, of course, n= -3= 26- 3= 23 (mod 26).

As a check, 23(17)= 442= 391= 390+ 1= 15(26)+ 1= 1 (mod 26).

1/17= 23 (mod 26) so x= 13/17= 13(23)= 299= 11(26)+ 13= 13 (mod 26).

1. What is a "system of equations using modulo"?

A system of equations using modulo refers to a set of equations that use the mathematical operation of "modulo" or "modulus", denoted by the symbol "%". This operation calculates the remainder when one number is divided by another, and is commonly used in computer programming and cryptography.

2. How is modulo used in solving a system of equations?

Modulo can be used to solve a system of equations by finding values for the variables that satisfy all of the equations while also satisfying a given modulus. This is useful for finding solutions to equations that involve cyclical patterns, such as finding the day of the week for a certain date or finding solutions to congruence equations.

3. What are some real-life applications of systems of equations using modulo?

Systems of equations using modulo have various real-life applications, including in cryptography for secure communication and authentication, in computer graphics for creating repeating patterns or animations, and in scheduling for optimizing work shifts and rotations.

4. Can a system of equations using modulo have multiple solutions?

Yes, a system of equations using modulo can have multiple solutions. In fact, most systems of equations using modulo will have more than one solution due to the cyclical nature of the modulo operation. It is important to check all possible solutions and consider the context of the problem to determine which solution is the most appropriate.

5. Are there any limitations when using modulo in systems of equations?

While modulo can be a useful tool in solving systems of equations, there are some limitations to keep in mind. Modulo cannot be used when dividing by 0, and it may also not be applicable for equations with non-integer solutions. Additionally, it is important to be aware of possible errors when using large numbers or negative numbers in modulo operations.

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