0
votes
0answers
29 views

Isomorphic matrix groups over rings

I've thinking about this problem for the last couple days and I can't get anywhere. I would really appreciate some help. Is it true that, a) $\operatorname{SL}_n(\mathbb{Z}/2013\mathbb{Z})\cong ...
6
votes
1answer
61 views

Matrix restoring (modulo n)

Let $m,n\geqslant 2$, and $A\in \mathcal{M}_n(\mathbb Z)$ such that $\det A \equiv 1 \pmod m$. Does it (necessarily) exist $M\in \mathrm{GL}_n(\mathbb Z)$ such that $A\equiv M \pmod m$?
0
votes
0answers
61 views

Matrix Induction proofs using $\rm mod$.

Please could someone advise me on where to even start with this question. Thanks in advance Let $A$ be an $n\times n$ matrix, for some $n\geqslant 1$. Given any prime number $p\gt1$, write $A_p$ for ...
0
votes
1answer
20 views

Matrix double modulo multiplication to get identity

I have to multiply to matrices A and B which can consist of numbers 0,2,3,4,5,6 to get an identity matrix, however multiplication happens with moduli after every ...
0
votes
1answer
63 views

Does Fermat's Little Theorem apply to matrices?

I'm working on a problem involving applying FLT to matrices, so any information about how to do this or prove this is true would be helpful. I've been doing some research and experimenting a little, ...
0
votes
0answers
209 views

Kronecker-Capelli Theorem for system of congruences

Let $p$ be some prime. Given a system of linear congruences, \begin{align} m_1 x + n_1 y &\equiv c_1 \quad (mod\, p)\\ m_2 x + n_2 y &\equiv c_2 \quad (mod\,p)\\ \ldots \\ m_d x + n_d y ...
1
vote
2answers
59 views

Show that $M$ is a submonoid of the group of $2\times 2$ matrices of integers mod $13$.

Define $$M=\left\{\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}\in(\Bbb Z/13)_{22}:a_{12}=0\right\}\;.$$ (This is the set of $2\times 2$ matrices with integers mod $13$ with the ...
1
vote
1answer
40 views

Determining order of matrices in $GL_2(\mathbb{F}_7)$

I need to determine the orders of the following matrices in the group $GL_2(\mathbb{F}_7)$: $$\begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix} \text{ and } \begin{pmatrix} 2 & 0\\ 0 & 1 ...
2
votes
2answers
88 views

Solving a system of equation modul0 5

Consider the system of linear equations $$\begin{pmatrix} 6 & -3\\ 2 & 6 \end{pmatrix}\begin{pmatrix} x_1\\ x_2 \end{pmatrix}=\begin{pmatrix} 3\\ 1 \end{pmatrix} $$ a) Solve the system in ...
8
votes
1answer
92 views

Finding the generators of a subgroup of $\mathrm{SL}_2(\mathbb Z)$

I am trying to solve the following problem: Let $T_{ij}(c)\in\mathrm{SL}_2(\mathbb Z)\ (i\neq j)$ be the elementary matrix which represents the elementary row operation of adding the $j$-th row ...
1
vote
1answer
65 views

$(A^x) \mathbin\% p = (A^{x \mathbin\% (p - 1)}) % p$ if $p$ is prime. Is this true when $A$ is a matrix?

$(A^x) \mathbin\% p = (A^{x \mathbin\% (p - 1)}) % p$ if $p$ is prime. Is this property true when $A$ is a matrix? Suppose $$A=\begin{pmatrix} 1 &0 &1\\ 1 &0 &0\\ 0 &1 &0 ...
0
votes
1answer
385 views

Matrix Multiplication under a Modulo

Let $a$ and $b$ be different non-singular square matrices (same dimenstions) where all values are between 0 - 15 Let $c$ = $a.b$ mod 16 (all values in matrix are changed to mod 16) Will ...
1
vote
0answers
46 views

Why the following observations regarding lattices hold?

The following is an excerpt of a recent paper on lattice cryptography: Let $n$ and $q$ be integers [...], and let $\beta > 0$ . Given a uniformly random matrix $A \in \mathbb Z^{n \times m}_q$ ...
0
votes
1answer
47 views

Matrix inverses over finite fields with composite moduli

I know that over a field $F$, a matrix is invertible if and only if its determinant is nonzero. And I understand why this is true, at least in the case where the field is $\mathbb{R}$. But I do not ...
1
vote
1answer
104 views

Matrix inversion help

If $g(n)$ is an integer functions periodic in $a$ And $\phi(a)$ is eulers totient function And $[r_1,r_2,r_3,...r_{\phi(a)}]$ are the postive integers less then $a$ coprime to $a$ With ...
0
votes
1answer
49 views

Can this function with modulo and truncated division be simplifed?

Can this function with modulo and truncated division (DIV) be simplifed? f(x)=(x%c)*r+DIV(x,c)%r Basically, I use this ...
1
vote
2answers
1k views

Inverse of a matrix mod 26

I'm trying to find the inverse of the matrix $\begin{bmatrix} 4 &8 \\ 5 &7 \end{bmatrix} \mod 26$. However the determinant of this matrix is 14 so I cannot use Cramer's rule and each time ...
4
votes
0answers
53 views

Prove that there are $736$ $2 \times 2$ matrices ($A$) where $A=A^{-1}$ [duplicate]

I'm doing some assignments to teach myself cryptology. I am still at the introductory cryptology level, where a lot of it is discrete mathematics, so I believe - and hope - that it is a somewhat ...
11
votes
5answers
323 views

Why are there$ 736$ $2\times 2$ matrices $(M)$ over $\mathbb{Z}_{26}$ for which it holds that $M=M^{-1}$?

I'm currently trying to introduce myself to cryptography. I'm reading about the Hill Cipher currently in the book Applied Abstract Algebra. The Hill Cipher uses an invertible matrix $M$ for ...
2
votes
0answers
121 views

A system of linear equations in integer squares - solvable?

I consider this more a "recreational math" problem, possibly lacking a solution (because it stems from the question of "magic square of squares") or simply intractable with reasonable effort. ...
1
vote
1answer
172 views

linear equations - under modulo

I'm having difficulties proceeding with this problem: We have the following linear equations: $$\begin{array} 1x + 2y + 2z = 0 \\ 3x – 2y + 2z =1\\ 2x + y + z =3\end{array}$$ ...
2
votes
2answers
93 views

Counting the number of matrices which cause collision

Let $m,n \in \mathbb{N}$, and $q$ be a prime number. Let $\mathbf{A} \in \mathbb{Z}^{m \times n}_q$ be a matrix. In the following, assume that all additions and multiplications are performed modulo ...
39
votes
14answers
3k views

'Linux' math program with interactive terminal?

Are there any open source math programs out there that have an interactive terminal and that work on linux? So for example you could enter two matrices and specify an operation such as multiply and ...
0
votes
2answers
162 views

Need help with finding matrix inverse in $\mathbb{Z}/26\mathbb{Z}$

I am trying to learn the Hill Cipher and I am facing difficulties understanding how to find the inverse of a matrix in Modulo 26. What I've learnt so far is that I need to apply elementary row ...
1
vote
2answers
258 views

What is the meaning of symmetric modulo?

So I was reading this: http://rjlipton.wordpress.com/2009/04/01/a-new-factoring-algorithm/ and found it saying "symmetic modulo" So first off: what is " a -special matrix provided the following ...
1
vote
0answers
99 views

How do you find small coefficients that satisfy a particular modular equation

Let's say $p=16301$. How do I best find sets of small values for $a$, $b$ and $c$ for an equation like $$a p^3+b p^2+c p=11263 \mod\ 2^{16}.$$ I can use the ...
2
votes
3answers
1k views

I don't understand why the inverse is this?

my question is related to matrix inverting and Hill cipher(you don't have to know what it is to help me) My teacher gave me an example. First we have a matrix (the key matrix) that multiplied by a ...