# Matrix exponent $A^n$ in modular arithmetic $\mathbb{Z}/10\mathbb{Z}$

Let $$A=\begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}, A_{ij} \in \mathbb{Z}/{10\mathbb{Z}}$$.

Determine how many different $$A^n$$ there are and which of them are equal to $$A$$.

My attempt: Simple multiplication doesn't yield any results, so I assume I'm supposed to apply some algebraic property of $$\mathbb{Z}/10\mathbb{Z}$$ but I'm not sure which one would be helpful in this case. Maybe something about it being a cyclic group. Any hints would be helpful!

Edit:

The matrix $$B = A^2 = \begin{bmatrix}0 & 6 & 2 \\ 6 & 1 & 6 \\ 2 & 6 &0 \end{bmatrix}$$ is symmetric and it seems that $$B^5 = B$$, is there anything that would justify this and maybe enable discovery without direct calculation?

• I am somewhat unfamiliar with linear algebra over something that's not a field, but presumably, the Cayley-Hanilton theorem still holds. – Arthur Dec 22 '20 at 13:04
• Does it help if you do your calculation $\pmod 2$ and separately $\pmod 5$? – Stinking Bishop Dec 22 '20 at 13:05
• @Collapse Because that lets you express $A^3$ in terms of $A^2,A$ and $I$. Which again lets you express $A^4$ in terms of $A^2,A$ and $I$. And so on. Which is to say, you can set up a simple recursion for $A^n$ that should be very solvable. – Arthur Dec 22 '20 at 17:41
• @Collapse I was thinking that the problem may be more easily tractable $\pmod 5$. For example, $\pmod 5$ you have $A^3=3A$, which implies that $A^5=3^2A=4A, A^7=3^3A=2A, A^9=3^4A=A$, so really all the powers of $A\pmod 5$ are $A,A^2,\ldots A^8$. Similarly, $A^3=A^2$ and then $A^4=A^5=\ldots=A^2$, so the only powers of $A\pmod 2$ are $A, A^2$. Now use Chinese Remainder Theorem to conclude what the different powers are $\pmod{10}$. – Stinking Bishop Dec 22 '20 at 17:46
• (Cont'd): In fact, (I used Cayley-Hamilton as suggested by @Arthur): the characteristic polynomial of the matrix over $\mathbb Q$ is $-\lambda^3+15\lambda^2+18\lambda$, which means that $A^3-15A^2-18A=0$ - but this is then also valid $\pmod 5$, where $15=0, 18=3$, i.e. $A^3=3A$. – Stinking Bishop Dec 22 '20 at 17:54

This is a very interesting question.

EDIT 06.01.2021

Introducing what I call the equivalent 2x2-matrix appoach simplifies and unifies the task of finding $$q(d)$$ for all dimensions. The task now boils down to finding the order of an exponential congruence generalized to the 2x2 matrix.

EDIT 01.01.21

Explicit formulas for the mod(m) of $$\alpha$$ and $$\beta$$.

EDIT 30.12.20

Pursuing the second approach, in the case of even dimension $$d$$ we find a simple exponential congruence $$\beta^k \equiv 1 \text{ mod(m)}$$ ($$\beta$$ and $$m$$ are known integer functions of $$d$$) for which we need to find the smallest positive solution of the exponent $$k$$.

The general solution for this restricted case is still open (to me). Help from experts in number theory is greatly appreciated.

The more complicated case of odd $$d$$ has still to be done.

EDIT 27.12.20

Extension to two approaches to solve the generalized problem:

1. Brute force matrix multiplication (as in the original post)

2. Employing the characteristic polynom of the matrix and the Cayley-Hamilton theorem to derive a linear recurrence of the Fibonacci type

Generalized problem

Let $$a(d)$$ be a square matrix of dimension $$d$$ formed by consecutively filling in the numbers 1 through d^2 in lexicographic order.

The elements are hence

$$a_{ik}=i+d(k-1)\tag{1}$$

Now form the set $$s(d) = (a, a^2, a^3, ..., a^q)$$ of consecutive powers of the matrix $$a$$ taken modulo $$m=d^2+1$$, which we designate by $$[..]$$. The power $$q$$ is defined by the maximum value of $$q$$ for which all elements in the set $$s(q)$$ are different.

Task: find $$q(d)$$

Solution

0) Equivalent 2x2 matrix

This approach is a compact formulation of the results of my earlier studies documented before in this answer. It is valid for all dimensions $$d$$ and is based on the following theorem 1 which I provide here for the moment without proof.

Theorem 1

Consider the matrix

$$A(d) = \begin{bmatrix}\alpha(d) & 1 \\ \beta(d)&0 \end{bmatrix}\tag{0.1}$$

where

$$\alpha(d) = \frac{1}{2} d(d^2+1)\tag{0.2a}$$ $$\beta(d) = \frac{1}{12}d^3(d^2-1)\tag{0.2b}$$

and let

$$m=1+d^2\tag{0.3}$$

Then the number $$q(d)$$ of different powers of the original matrix $$a$$ of the OP defined in $$(1)$$ is given for $$d$$ even by the order $$k$$, and for $$d$$ odd by $$k+1$$ in the exponential congruence

$$A^{k+1} \equiv A \text{ (mod m)}\tag{0.4}$$

where the action of the module function on a matrix is defined as the action on its elements. Generalizing slightly the terminology of exponential congruences the order of a matrix $$A$$ modulo $$m$$ is defined as the smallest number $$k>0$$ for which $$(0.4)$$ holds.

1) Brute force calculation of the powers of the matrix of the OP

Let us start small.

For $$d=2$$ the first 9 powers of

$$a = \begin{bmatrix}1 & 2 \\ 3&4 \end{bmatrix}$$

mod 5 are

$$\begin{array}{cc} a = \begin{bmatrix}1 & 2 \\ 3&4 \end{bmatrix}, [a^2] = \begin{bmatrix}2 & 0 \\ 0&2 \end{bmatrix}, [a^3] = \begin{bmatrix}2 & 4 \\ 1&3 \end{bmatrix}, [a^4] = \begin{bmatrix}4 & 0 \\ 0&4 \end{bmatrix},\\ [a^5] = \begin{bmatrix}4 & 3 \\ 2&1 \end{bmatrix}, [a^6] = \begin{bmatrix}3 & 0 \\ 0&3 \end{bmatrix}, [a^7] = \begin{bmatrix}3 & 1 \\ 4&2 \end{bmatrix},\\ [a^8] = \begin{bmatrix}1 & 0 \\ 0&1 \end{bmatrix}, [a^9] = \begin{bmatrix}1 & 2 \\ 3&4 \end{bmatrix} \end{array} \tag{2}$$

We see that $$[a^9] = a$$ so that $$q(2)=8$$.

For $$d=3$$, the case of the OP, we have

$$\begin{array}{ccc} a= \begin{bmatrix}1 & 2&3 \\ 4&5&6\\7&8&9 \end{bmatrix}, [a^2] = \begin{bmatrix}0 & 6&2 \\ 6&1&6\\2&6&0 \end{bmatrix}, [a^3] = \begin{bmatrix}8&6&4 \\ 2&5&8\\6&4&2 \end{bmatrix},\\ [a^4] = \begin{bmatrix}0&8&6 \\ 8&3&8\\6&8&0 \end{bmatrix}, [a^5] = \begin{bmatrix}4&8&2 \\ 6&5&4\\8&2&6 \end{bmatrix}, [a^6] = \begin{bmatrix}0&4&8\\9&4&9\\8&4&0 \end{bmatrix},\\ [a^7] = \begin{bmatrix}2&4&6\\8&5&2\\4&6&8\end{bmatrix}, [a^8] = \begin{bmatrix}0&2&4\\2&7&2\\4&2&0\end{bmatrix}, [a^9] = \begin{bmatrix}6&2&8\\4&5&6\\2&8&4\end{bmatrix},\\ [a^{10}] = \begin{bmatrix}0&6&2\\6&1&6\\2&6&0\end{bmatrix} \end{array} \tag{3}$$

Interestingly, for $$d=3$$, the case of the OP, there is no power $$k\gt 1$$ for which $$[a^k] = a$$ but we have $$[a^{10}] = [a^2]$$, hence $$q(3)=9$$.

Now we let mathematica work out the first 20 values of $$q(d)$$:

$$q(d) =\\ \{1,8,9,32,7,2,41,24,81,40,\\31,56,33,56,225,512,57,120,61,800\}\tag{4}$$

2) Employing the characteristic polynomial of the matrix and reducig the problem to a linear recurrence relation of second order

The characteristic polynomial of a matrix $$A$$ is defined by

$$p(A,\lambda) = \left| \lambda I - A\right|\tag{5}$$

and the characteristic equation $$p(A,\lambda)=0$$ gives the eigenvalues $$\lambda$$ of $$A$$ as roots.

For our matrix $$a$$ defined by $$(1)$$ the list of the first ten polynomials for $$d=1..10$$ in the format $$\{d,p\}$$ is

$$\begin{array}{c} \{1,\lambda -1\} \\ \left\{2,\lambda ^2-5 \lambda -2\right\} \\ \left\{3,\lambda ^3-15 \lambda ^2-18 \lambda \right\} \\ \left\{4,\lambda ^4-34 \lambda ^3-80 \lambda ^2\right\} \\ \left\{5,\lambda ^5-65 \lambda ^4-250 \lambda ^3\right\} \\ \left\{6,\lambda ^6-111 \lambda ^5-630 \lambda ^4\right\} \\ \left\{7,\lambda ^7-175 \lambda ^6-1372 \lambda ^5\right\} \\ \left\{8,\lambda ^8-260 \lambda ^7-2688 \lambda ^6\right\} \\ \left\{9,\lambda ^9-369 \lambda ^8-4860 \lambda ^7\right\} \\ \left\{10,\lambda ^{10}-505 \lambda ^9-8250 \lambda ^8\right\} \\ \end{array} \tag{6}$$

The general form of $$p(a,d)$$ can be written as

$$p(a,d) = \lambda ^d-\alpha ( d) \lambda ^{d-1}-\beta ( d) \lambda ^{d-2}\tag{7}$$

where

$$\alpha (d)=\frac{1}{2} d \left(d^2+1\right)\tag{8a}$$

$$\beta (d)=\frac{1}{12} d^3 \left(d^2-1\right)\tag{8b}$$

Now the theorem of Cayley and Hamilton states that a matrix obeys its own characteristic equation.

For example from $$(6$$) we get for $$d=2$$ the equation

$$a^2 = 5a + 2\tag{9}$$

where we have dropped writing down the unit matrix $$I$$.

This equation allows to write higher powers of $$a$$ in terms of a linear combination of lower powers. Indeed, if we also take into account that all terms should be taken mod $$d^2+1=5$$ for $$d=2$$, the elements of the set $$s$$ are can be calculated as follows

$$a, \\a^2 \to 5a+2\equiv 2 \text{ (mod 5)}, \\a^3=5a^2+2a \equiv_{5} 2a \\a^4 = 2a^2\to 2\times 2= 4\\a^5 = 4a\\a^6 \equiv_{5} 4a^2\to 4 \times 2 = 8\equiv_{5} 3\\a^7=3a\\ a^8 = 3a^2 \to 3 \times 2 = 6 \equiv_{5} 1$$

Hence

$$s(d=2) = \{a, 2, 2a, 4, 4a, 3, 3a, 1\}$$

which has $$8$$ different elements, hence $$q(2)=8$$.

The generalization is not difficult. We have to employ $$(7)$$ to get the general power of $$a$$.

We are going to do this by solving the corresponding difference equation

$$c_{k} = \alpha c_{k-1} + \beta c_{k-2}\tag{10}$$

with the initial conditions

$$c_1 = a, c_2 = a^2\tag{11}$$

$$(10)$$ is a linear difference equation of the Fibonacci type which can be easily solved by standard methods.

The result is with two constants $$u$$ and $$v$$

$$c(k) = u \; . 2^{-k} \left(\alpha -\sqrt{\alpha ^2+4 \beta }\right)^k\\+v\;. 2^{-k} \left(\alpha +\sqrt{\alpha^2+4 \beta }\right)^k\tag{12}$$

Modular arithmetic

We pause here for a moment to study the modular arithmetic of the function $$\alpha(d)$$ and $$\beta(d)$$, i.e. their remainder mod $$m=(1+d^2)$$.

For $$\alpha$$ we have

$$\alpha(d \text{ even})\equiv 0\tag{13a}$$

$$\alpha(d \text{ odd})\equiv_{m} \frac{1}{2}(1+d^2)\tag{13b}$$

For $$\beta$$ things are more complicated.

The first few terms for $$\beta(d)$$ mod($$m$$) are

$$\beta(d) |_{d=1}^{d=20} \equiv_{m} \\ \{0, 2, 8, 12, 16, 1, 22, 23, 22, 69, 12, 2, 158, 68, 172, 174, 172, 3, 154, 137\}\tag{14a}$$

We can analyse the structure of this sequence by plotting it.

For even d we get

and for odd d

We see that there are "hidden" sub-sequences. Specifically, three and six sub-sequences for even and odd d, respectively.

Closer inspection shows that we have explicit formulas for the sequences as follows

$$\beta(d \text{ even))}\equiv_{m} \left\{ \begin{array}{ll} \frac{d}{6} & \;\; d \equiv_{6} 0\\ 12 \left(\frac{d-2}{6}\right)^2+\frac{9 (d-2)}{6}+2 & \;\; d \equiv_{6} 2\\ 24 \left(\frac{d-4}{6}\right)^2+\frac{33 (d-4)}{6}+12 & \;\; d \equiv_{6} 4\\ \end{array} \right. \tag{14b}$$

$$\beta(d \text{ odd))}\equiv_{m} \left\{ \begin{array}{ll} 132 \left(\frac{d-1}{12}\right)^2+\frac{24 (d-1)}{12}+2 & \;\; d \equiv_{12} 1\\ 108 \left(\frac{d-3}{12}\right)^2+\frac{56 (d-3)}{12}+8 & \;\; d \equiv_{12} 3\\ 84 \left(\frac{d+7}{12}\right)^2-\frac{96 (d+7)}{12}+28 & \;\; d \equiv_{12} 5\\ 60 \left(\frac{d+5}{12}\right)^2-\frac{48 (d+5)}{12}+10 & \;\; d \equiv_{12} 7\\ 36 \left(\frac{d+3}{12}\right)^2-\frac{16 (d+3)}{12}+2 & \;\; d \equiv_{12} 9\\ 12 \left(\frac{d+1}{12}\right)^2 & \;\; d \equiv_{12} 11\\ \end{array} \right. \tag{14c}$$

Now that we have explicit formulas for the mod(m) of $$\alpha$$ and $$\beta$$ we can dive deeper.

Let us start with the case of even $$d$$ where $$\alpha\equiv_{m}=0$$ and the characteristic equation is simply

$$a^3 = \beta(d) a\tag{15}$$

(to do: the next steps have to be revised in view of $$(14b)$$ and $$(14c)$$).

Explicitly the first 10 equations are in the format {d, m, equn} are

$$\begin{array}{c} \left\{2,5,a^3\equiv 2 a\right\} \\ \left\{4,17,a^3\equiv 12 a\right\} \\ \left\{6,37,a^3\equiv a\right\} \\ \left\{8,65,a^3\equiv 23 a\right\} \\ \left\{10,101,a^3\equiv 69 a\right\} \\ \left\{12,145,a^3\equiv 2 a\right\} \\ \left\{14,197,a^3\equiv 68 a\right\} \\ \left\{16,257,a^3\equiv 174 a\right\} \\ \left\{18,325,a^3\equiv 3 a\right\} \\ \left\{20,401,a^3\equiv 137 a\right\} \\ \end{array} \tag{16}$$

All sets $$s$$ start with $$\{a,a^2,a^3\}$$. Now $$a^3$$ has to be replaced from the corresponding by $$\beta(d) a$$, the next element is $$\beta a^2$$, and so on.

Hence

$$s = \{a,a^2, \beta a, \beta a^2, \beta^2 a, \beta^2 a^2, ...,\beta^k a, \beta^k a^2 \}\tag{17}$$

The last two elements of s are a repetition of the first two if

$$\beta(d \text{ even})^k \equiv 1 \text{ mod}(m)\tag{18a}$$

Remember that $$m=1+d^2$$ and that $$\beta$$ is given by $$(8b)$$.

Let $$k_{min}$$ be the smallest positive value which solves $$(18a)$$. The number $$k_{min}$$ is called the order of $$\beta$$ modulo $$m$$. Then the number of elements of the set s is given by

$$q(d) = |s| = 2 k_{min}\tag{18b}$$

Our task is hence reduced for even $$d$$ to the problem of finding the order of $$\beta(d)$$ modulo $$1+d^2$$.

Concerning its general solution we have to look up the theory of exponential congruences. Nice examples are Solution to exponential congruence and https://www.youtube.com/watch?v=oCgWwB465p8.

However, an upper limit of $$k$$ can be found using the little Fermat theorem in the formuation of Euler which states that if $$m$$ and $$\beta$$ have no common divisors (which is the case in our examples) then

$$k = k_{\phi} = \phi(m)\tag{19}$$

solves $$(18a)$$. Here $$\phi(m)$$ is Euler's totient function. Unfortunately, in most cases $$k_{\phi}$$ is not the smallest solution of $$(18a)$$.

Indeed, for d=8 we have m=65, $$\beta=23$$, $$\phi(65) = 48$$ but we have already $$23^{12}\equiv 1 \text{ mod 65}$$

It is helpful to notice that the order of $$\beta$$ modulo $$m$$ must be a divisor of $$\phi(m)$$. Hence we have reduced the possible number of orders. In the case of prime $$m$$ we have $$\phi(m) = m-1=d^2$$.

An even more effective procedure to find the order is described here https://math.stackexchange.com/a/165728/198592.

Discussion

For the specific problem of the OP which has $$d=3$$ the answer is 9. There are 9 different matrices.

Two values appear twice ($$q(12)=q(14)=56$$.

The case $$d=6$$ is remarkable since here already $$[a^3] = a$$

I have not found a general formula for $$q(d)$$. This might be difficult since the sequence is not contained in https://oeis.org/.

Matrix multiplication and taking the module

We can verify that for two matrices $$a$$ and $$b$$ with integer elements we have the relation

$$[ a.b ] = [[a].[b]]\tag{5}$$

Hence the two operations can be interchanged.

Group properties?

The set $$s$$ does not form a group for the simple reason that the matrix $$a$$ is singular and hence has no inverse.

The even powers have an inverse but in general have non integer elements, and also do not generate a unit element.

Rather, for each $$d$$, the set $$s$$ is an algebra with a multiplication rule defined by the characteristic equation.