EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 1, 2024, 11-29
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Determinants of Arrowhead Matrices over Finite
Commutative Chain Rings

Somphong Jitman1,∗, Pornrudee Modjam1

1 Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom
73000, Thailand

Abstract. Arrowhead matrices have attracted attention due to their rich algebraic structures and
numerous applications. In this paper, we focus on the enumeration of n × n arrowhead matrices
with prescribed determinant over a finite field Fq and over a finite commutative chain ring R. The
number of n×n arrowhead matrices over Fq of a fixed determinant a is determined for all positive
integers n and for all elements a ∈ Fq. As applications, this result is used in the enumeration of
n × n non-singular arrowhead matrices with prescribed determinant over R. Subsequently, some
bounds on the number of n × n singular arrowhead matrices over R of a fixed determinant are
given. Finally, some open problems are presented.

2020 Mathematics Subject Classifications: 11C20, 15B33

Key Words and Phrases: Arrowhead matrices, Determinants, Finite fields, Finite commutative
chain rings, Enumeration

1. Introduction

Matrices and their determinants have been known and extensively studied for their
nice properties and wide applications (see, for example, [2], [9], and [10]). Singularity of
matrices is useful in applications (see, for example, [2] and [11]). The number of n × n
singular (resp., nonsingular) matrices over a finite field Fq has been determined in [13].
As a generalization of a prime field Zp, the number of n × n matrices over Zm of a fixed
determinant has been first studied in [1]. An alternative study of the problem in [1] has
been given in [10] using a different and simpler approach. A finite commutative chain ring
(FCCR) and a principal ideal ring are generalizations of the rings Zp and Zm that are
useful in applications such as coding theory and cryptography. In [3], the techniques in
[10] have been extended to matrices over FCCRs and principal ideal rings. Precisely, the
number of n × n matrices over FCCRs and principal ideal rings of a fixed determinant
has been completely determined. Diagonal matrices are interesting subfamilies of the ones

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.4983

Email addresses: sjitman@gmail.com (S. Jitman), pornrudee.ole@gmail.com (P. Modjam)

https://www.ejpam.com 11 © 2024 EJPAM All rights reserved.



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 12

in [3]. The enumeration of diagonal matrices over FCCRs of a fixed determinant are
presented in [8] and applied in the study of the determinant of some circulant matrices
over FCCRs.

For a commutative ring R and a positive integer n, an n × n arrowhead matrix over
R is defined to be a square matrix containing zeros in all entries except for the first row,
first column, and main diagonal. Precisely, the arrowhead matrix is in the form of

A =



∗ ∗ ∗ ∗ · · · ∗
∗ ∗ 0 0 · · · 0
∗ 0 ∗ 0 · · · 0
∗ 0 0 ∗ · · · 0
...

...
...

...
. . .

...
∗ 0 0 0 · · · ∗


,

where ∗’s are arbitrary elements in R and they are not necessarily the same. From the
definition, an arrowhead matrix is a generalization of a diagonal matrix over R. It is
easily seen that the 1 × 1 matrices, 2 × 2 matrices, and n × n diagonal matrices over R
are arrowhead matrices for all positive integers n. Some properties of arrowhead matrices
such as eigenvalues, eigenvectors, and inverses have been studied in [14], [15], and [16].
Arrowhead matrices have applications in various fields, e.g., wireless communications in
[15], eigenvalue decompositions of some matrices in [16], the study of directed multigraphs
and hub-directed multigraphs in[12], and the study of disordered quantum spins in [4].

As a generalization of [8], the enumeration of arrowhead matrices with prescribed
determinant over a FCCR is investigated in the following set up. For a FCCR R, let U(R)
denote the set of units in R and let Z(R) denote the set of zero-divisors in R. Let An(R)
denote the set of n× n arrowhead matrices over R. It is not difficult to see that An(R) is
a group under addition and

|An(R)| = |R|3n−2. (1)

An n × n matrix A over R is said to be non-singular (or, invertible) if det(A) ∈ U(R).
Otherwise, A is called a singular matrix. Let

IAn(R) = {A ∈ An(R) | det(A) ∈ U(R)}

be the set of n× n non-singular arrowhead matrices over R. For each a ∈ R, let

An(R, a) = {A ∈ An(R) | det(A) = a}.

be the set of all n× n arrowhead matrices over R whose determinant is a. Clearly,

IAn(R) =
⋃

a∈U(R)

An(R, a)

is a disjoint union.



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 13

The main focus of this paper is the enumeration of n × n arrowhead matrices with
prescribed determinant over a finite field Fq and over a FCCR R. The paper is organized
as follows. The number |An(Fq, a)| of n × n arrowhead matrices over Fq of determinant
a is determined for all positive integers n and for all elements a ∈ Fq in Section 2. As
applications, these results are used in the enumeration of arrowhead matrices of a fixed
determinant over R in Section 3. The number of n × n non-singular arrowhead matrices
of a fixed determinant over R in Subsection 3.1. Subsequently, bounds on the number
of n × n singular arrowhead matrices over R of some fixed determinant are discussed in
Subsection 3.2. Some remarks and open problems are given in Section 4.

2. Determinants of Arrowhead Matrices over Fq

In this section, we focus on the enumeration of arrowhead matrices of a fixed determi-
nant over a finite field Fq. For an element a ∈ Fq, the formula for the number of n × n
arrowhead matrices over Fq of determinant a is given for all prime powers q and positive
integers n.

A recursive formula for the number |IAn(Fq)| of n×n non-singular arrowhead matrices
over Fq is given in Proposition 1. Later, an explicit formula for |IAn(Fq)| is established
in Theorem 1 based on Proposition 1.

Proposition 1. Let q be a prime power. Then

|IA1(Fq)| = q − 1

and

|IAn(Fq)| = q2n−3(q − 1)n + q2(q − 1)|IAn−1(Fq)|
for all integers n ≥ 2.

Proof. Clearly, |IA1(Fq)| = |Fq \ {0}| = q − 1. Let n ≥ 2 be an integer and let

A =



a11 a12 a13 · · · a1,n−1 a1n
a21 a22 0 · · · 0 0
a31 0 a33 · · · 0 0
...

...
...

. . .
...

...
an−1,1 0 0 · · · an−1,n−1 0
an1 0 0 · · · 0 ann


∈ IAn(Fq).

For each i ∈ {1, 2, . . . , n}, let Ri (resp., Ci) denote the ith row (resp, ith column) of
A. We consider the two cases.
Case 1: ann ̸= 0. Applying the elementary row operation R1 − a1nann

−1Rn → R1 and
the elementary column operation C1 − an1ann

−1Cn → C1, it follows that

A ∼


0

C
...
0

0 · · · 0 ann

 ,



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 14

where

C =


a11 − a1nan1ann

−1 a12 a13 · · · a1,n−1

a21 a22 0 · · · 0
a31 0 a33 · · · 0
...

...
...

. . .
...

an−1,1 0 0 · · · an−1,n−1

 .

Then det(A) = (−1)n+nann det(C) = ann det(C).
Let

S =




s11 s12 s13 · · · s1,n−1

s21 s22 0 · · · 0
s31 0 s33 · · · 0
...

...
...

. . .
...

sn−1,1 0 0 · · · sn−1,n−1

 ∈ An−1(Fq)

∣∣∣∣∣∣∣∣∣∣∣
det




s11 − a1nan1ann

−1 s12 s13 · · · s1,n−1

s21 s22 0 · · · 0
s31 0 s33 · · · 0
...

...
...

. . .
...

sn−1,1 0 0 · · · sn−1,n−1



 ̸= 0


.

It follows that 
s11 s12 s13 · · · s1,n−1

s21 s22 0 · · · 0
s31 0 s33 · · · 0
...

...
...

. . .
...

sn−1,1 0 0 · · · sn−1,n−1

 ∈ S

if and only if 
s11 − a1nan1ann

−1 s12 s13 · · · s1,n−1

s21 s22 0 · · · 0
s31 0 s33 · · · 0
...

...
...

. . .
...

sn−1,1 0 0 · · · sn−1,n−1

 ∈ IAn−1(Fq).

Consequently, we have |S| = |IAn−1(Fq)|. We note that 0 ̸= det(A) = ann det(C) if and
only if det(C) ̸= 0, or equivalently,

a11 a12 a13 · · · a1,n−1

a21 a22 0 · · · 0
a31 0 a33 · · · 0
...

...
...

. . .
...

an−1,1 0 0 · · · an−1,n−1

 ∈ S.



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 15

Hence, there are |S| = |IAn−1(Fq)| possibilities for C. The number of choices of a1n and
an1 are q2 and the number of choices for ann is q − 1. Hence, the number of arrowhead
matrices A in IAn(Fq) is

q2(q − 1)|IAn−1(Fq)|.

Case 2: ann = 0. Since det(A) ̸= 0, we have a1n ̸= 0 and an1 ̸= 0. Applying the
elementary row operation Ri − ai1an1

−1Rn → Ri for all i ∈ {1, 2, . . . , n − 1} and the
elementary column operation C1 ↔ Cn, we have

A ∼



a1n a12 a13 · · · a1,n−1 0
0 a22 0 · · · 0 0
0 0 a33 · · · 0 0
...

...
...

. . .
...

...
0 0 0 · · · an−1,n−1 0
0 0 0 · · · 0 an1


=: A′.

Since det(A′) = −det(A) ̸= 0 if and only if a1n, a22, . . . , an−1,n−1, an1 are non-zero, the
number of (a1n, a22, a33, . . . , an−1,n−1, an1) is (q−1)n, the number of (a12, a13, a14, . . . , a1,n−1)
is qn−1, and the number of (a21, a31, a41, . . . , an−2,1) is q

n−2 In this case, the number of A
in IAn(Fq) is

q2n−3(q − 1)n.

From the two cases, it can be deduced that

|IAn(Fq)| = q2n−3(q − 1)n + q2(q − 1)|IAn−1(Fq)|

as desired. ■
An explicit expression for the number |IAn(Fq)| can be derived using the recursive

formula given in Proposition 1 and the principle of mathematical induction.

Theorem 1. Let q be a prime power. Then

|IAn(Fq)| = q2n−3(q − 1)n(q + (n− 1))

for all positive integers n.

Proof. For n = 1, we have

|IA1(Fq)| = q − 1 = q2(1)−3(q − 1)1(q + (1− 1)).

Let k ≥ 2 be an integer. Assume that

|IAk−1(Fq)| = q2(k−1)−3(q − 1)k−1(q + ((k − 1)− 1)).

Using the recurrent relation given in Proposition 1, we have

|IAk(Fq)| = q2k−3(q − 1)k + q2(q − 1)|IAk−1(Fq)|
= q2k−3(q − 1)k + q2(q − 1)(q2(k−1)−3(q − 1)k−1(q + ((k − 1)− 1)))



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 16

= q2k−3(q − 1)k + q2k−3(q − 1)k(q + (k − 2))

= q2k−3(q − 1)k(q + (k − 1)).

Therefore, it follows that

|IAn(Fq)| = q2n−3(q − 1)n(q + (n− 1))

for all positive integers n. ■
In the following proposition, a relation between |An(Fq, 1)| and |An(Fq, a)| for all

a ∈ Fq \ {0} is key to study the enumeration of |An(Fq, a)| in Corollary 1.

Proposition 2. Let q a prime power and let n be a positive integer. Then

|An(Fq, 1)| = |An(Fq, a)|

for all a ∈ Fq \ {0}.

Proof. Let a ∈ Fq \ {0} and let f : An(Fq, 1) → An(Fq, a) be defined by

f(A) = diag(a, 1, 1, . . . , 1)A.

Let

A =


a11 a12 a13 · · · a1n
a21 a22 0 · · · 0
a31 0 a33 · · · 0
...

...
...

. . .
...

an1 0 0 · · · ann

 ∈ An(Fq, 1).

Then det(A) = 1,

f(A) = diag(a, 1, 1, . . . , 1)A =


aa11 aa12 aa13 · · · aa1n
a21 a22 0 · · · 0
a31 0 a33 · · · 0
...

...
...

. . .
...

an1 0 0 · · · ann

 ∈ An(Fq), (2)

and

det(f(A)) = det(diag(a, 1, 1, . . . , 1)A) = det(diag(a, 1, 1, . . . , 1)) · det(A) = a · 1 = a.

Hence, f(A) ∈ An(Fq, a). Since diag(a, 1, 1, . . . , 1) is invertible, we have that f is injective.
Let X ∈ An(Fq, a) and let A = diag(a−1, 1, 1, . . . , 1)X. Then we have A ∈ An(Fq) and

det(A) = det(diag(a−1, 1, 1, . . . , 1)X) = a−1 · a = 1. It follows that A ∈ An(Fq, 1) and

f(A) = f(diag(a−1, 1, 1, . . . , 1)X) = diag(a, 1, 1, . . . , 1)diag(a−1, 1, 1, . . . , 1)X = X.

Consequently, f is surjective.



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 17

It follows that f is a bijection from An(Fq, 1) onto An(Fq, a), and hence, |An(Fq, 1)| =
|An(Fq, a)|. ■

From Proposition 2, we have

|An(Fq, a)| = |An(Fq, 1)| = |An(Fq, b)|

for all a, b ∈ Fq \ {0}.
Based on Theorem 1 and Proposition 2, the next corollary can be derived.

Corollary 1. Let q be a prime power and let n be positive integer. Then

|An(Fq, a)| = q2n−3(q − 1)n−1(q + (n− 1))

for all a ∈ Fq \ {0}.

Proof. From Proposition 2, it follows that |An(Fq, a)| = |An(Fq, 1)| for all a ∈ Fq \ {0}.
Since

IAn(Fq) =
⋃

a∈Fq\{0}

An(Fq, a)

is a disjoint union and |Fq \ {0}| = q − 1, it follows that

|IAn(Fq)| = |Fq \ {0}||An(Fq, 1)| = (q − 1)|An(Fq, 1)|.

By Theorem 1 and Proposition 2, we have

|An(Fq, a)| = |An(Fq, 1)|

=
|IAn(Fq)|
q − 1

=
q2n−3(q − 1)n(q + (n− 1))

q − 1

= q2n−3(q − 1)n−1(q + (n− 1)).

This completes the proof. ■
We note that |An(Fq)| = q3n−2 and

|IAn(Fq)| = q2n−3(q − 1)n(q + (n− 1))

given in (1) and Theorem 1. The number |An(Fq, 0)| = |An(Fq)| − |IAn(Fq)| of n × n
singular arrowhead matrices over Fq follows in the next corollary.

Corollary 2. Let q be a prime power. Then

|An(Fq, 0)| = q3n−2 − q2n−3(q − 1)n(q + (n− 1))

for all positive integers n.



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 18

3. Determinants of Arrowhead Matrices over FCCRs

In this section, the enumeration of n×n arrowhead matrices with prescribed determi-
nant over R is discussed. The number of n × n non-singular (resp., singular) arrowhead
matrices over R is presented. For non-singular arrowhead matrices, the number of n×n ar-
rowhead matrices over R with a given determinant is established. For singular arrowhead
matrices, bounds on the number of n × n arrowhead matrices with a fixed determinant
over R are presented in some cases.

To be self-contained, a brief information of a FCCR is recalled. The reader may
refer to [5], [6], and [7] for more details. A ring R with identity 1 ̸= 0 is called a finite
commutative chain ring (FCCR) if it is finite, commutative, and its ideals are linearly
ordered by inclusion. Let R be a FCCR whose maximal ideal is generated by γ. Then the
ideals in R are of the form

R ⊋ γR ⊋ γ2R ⊋ · · · ⊋ γe−1R ⊋ γeR = {0},

for some positive integer e. The smallest positive integer e such that γe = 0 is called the
nilpotency index of R. The quotient ring R/γR is a finite field and it is referred to as the
residue field of R. From [6] and [7], useful properties of a FCCR (cf. [3]) are summarized
in the next lemma.

Lemma 1. Let R be a FCCR of nilpotency index e and let γ be a generator of its max-
imal ideal. Let V ⊆ R be a set of representatives for the equivalence classes of R under
congruence modulo γ. Assume that the residue field R/⟨γ⟩ ∼= Fq for some prime power q.
Then the following statements hold.

1) For each r ∈ R, there exist unique a0, a1, . . . ae−1 ∈ V such that

r = a0 + a1γ + · · ·+ ae−1γ
e−1.

2) |V | = q.

3) |γjR| = qe−j for all 0 ≤ j ≤ e.

4) U(R) = {a+ γb | a ∈ V \ {0} and b ∈ R}.

5) |U(R)| = (q − 1)qe−1.

6) For each 0 ≤ i ≤ e, R/γiR is a FCCR of nilpotency index i and residue field Fq.

3.1. Non-Singular Arrowhead Matrices over FCCRs

First, the number of n × n non-singular arrowhead matrices over a FCCRs R is pre-
sented. Then it is followed by the number of n × n arrowhead matrices over R with
prescribed determinant in U(R).

An explicit formula for the number |IAn(R)| of n × n non-singular matrices is given
in the following theorem.



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 19

Theorem 2. Let R be a FCCR with residue field Fq and nilpotency index e. Then

|IAn(R)| = qe(3n−2)−(n+1)(q − 1)n(q + (n− 1))

for all positive integers n.

Proof. Let γ be a generator of the maximal ideal of R and let φ : R → Fq be the ring
homomorphism defined by a 7→ a + ⟨γ⟩. By considering An(R) and An(Fq) as additive
groups, let ϕ : An(R) → An(Fq) be the group homomorphism defined by

A = [aij ] 7→ [φ(aij)].

It is not difficult to see that ϕ is a surjective homomorphism. By the First Isomorphism
Theorem for groups, it follows that An(Fq) ∼= An(R)/ ker(ϕ). Hence,

| ker(ϕ)| = |An(R)|
|An(Fq)|

=
qe(3n−2)

q3n−2
= q(e−1)(3n−2).

For A ∈ An(R), we have det(ϕ(A)) = φ(det(A)) which implies that det(A) is a unit in R
if and only if det(ϕ(A)) ̸= 0 in Fq. Equivalently, A is invertible over R if and only if ϕ(A)
is invertible over Fq. Then the restriction map ϕ|IAn(R) : IAn(R) → IAn(Fq) is surjective
and it is | ker(ϕ)| to one map. From Theorem 1, we have

|IAn(Fq)| = q2n−3(q − 1)n(q + (n− 1)).

It follows that

|IAn(R)| = | ker(ϕ)||IAn(Fq)|
= q(e−1)(3n−2)|IAn(Fq)|
= q(e−1)(3n−2)q2n−3(q − 1)n(q + (n− 1))

= qe(3n−2)−(n+1)(q − 1)n(q + (n− 1))

as desired. ■
For each a ∈ U(R), the relation between |An(R, 1)| and |An(R, a)| in the following

proposition is key to determine the number |An(R, a)| in Corollary 3.

Proposition 3. Let R be a FCCR and let n be a positive integer. Then

|An(R, a)| = |An(R, 1)|

for all a ∈ U(R).

Proof. Let a ∈ U(R) and let θ : An(R, 1) → An(R, a) be the map defined by

θ(A) = diag(a, 1, 1, . . . , 1)A.



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 20

Using arguments similar to those in the proof of Proposition 2, it can be deduced that θ
is a bijection from An(R, 1) onto An(R, a). As desired, |An(R, a)| = |An(R, 1)|. ■

From Proposition 3, it follows that |An(R, a)| = |An(R, 1)| = |An(R, b)| for all units
a, b ∈ U(R). For a fixed unit a ∈ R, the number of n × n arrowhead matrices over R
whose determinant is a will be given later in Corollary 3.

Corollary 3. Let R be a FCCR with residue field Fq and nilpotency index e and let n be
a positive integer. Then

|An(R, a)| = q3e(n−1)−n(q − 1)n−1(q + (n− 1))

for all a ∈ U(R).

Proof. First, we note that IAn(R) is disjoint union of An(R, a) for all a ∈ U(R). Precisely,

IAn(R) =
⋃

a∈U(R)

An(R, a)

is a disjoint union. By Proposition 3, An(R, a) has the same number of elements as
An(R, 1), and hence,

|IAn(R)| =

∣∣∣∣∣∣
⋃

a∈U(R)

An(R, a)

∣∣∣∣∣∣
=

∑
a∈U(R)

|An(R, a)|

=
∑

a∈U(R)

|An(R, 1)|

= |U(R)||An(R, 1)|.

From Lemma 1, we have |U(R)| = (q − 1)qe−1. By Proposition 3, it can be deduced that

|An(R, a)| = |An(R, 1)|

=
|IAn(R)|
|U(R)|

=
qe(3n−2)−(n+1)(q − 1)n(q + (n− 1))

(q − 1)qe−1

= q3e(n−1)−n(q − 1)n−1(q + (n− 1)).

The proof is completed. ■

3.2. Singular Arrowhead Matrices over FCCRs

In this subsection, the enumeration of singular arrowhead matrices with prescribed
determinant over a FCCR R are studied. Unlike the previous subsection, only bounds on



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 21

the number of singular n× n arrowhead matrices over R with prescribed determinant are
given.

Since the number of n × n arrowhead matrices over R is qe(3n−2), the next corollary
follow immediately from Theorem 2.

Corollary 4. Let R be a FCCR with residue field Fq and nilpotency index e. Then the
number of n× n singular arrowhead matrices over R is

qe(3n−2)−(n+1)
(
qn+1 − (q − 1)n(q + (n− 1))

)
for all positive integers n.

3.2.1. Singular Arrowhead Matrices over FCCRs with Zero Determinant

A general recursive lower bound on the number of n× n arrowhead matrices over R with
zero determinant is given in the next proposition. For e = 2, a more specific bound is
derived in Corollary 5.

Proposition 4. Let R be a FCCR of nilpotency index e and residue field Fq. If γ is a
generator of the maximal ideal of R, then |A1(R, 0)| = 1 and

|An(R, 0)| ≥ (q − 1)q2(e−1)(qe+1 + 1)|An−1(R, 0)|+ q3n−4|An(R/γ
e−1R, 0 + γe−1R)|

for all integers n ≥ 2.

Proof. Clearly, |A1(R, 0)| = 1. Let n ≥ 2 be an integer and let

A =



a11 a12 a13 · · · a1,n−1 a1n
a21 a22 0 · · · 0 0
a31 0 a33 · · · 0 0
...

...
...

. . .
...

...
an−1,1 0 0 · · · an−1,n−1 0
an1 0 0 · · · 0 ann


∈ An(R, 0).

For convenience, for each i ∈ {1, 2, . . . , n}, denote by Ri (resp., Ci) the ith row (resp,
ith column) of A. We consider the following two cases.
Case 1: a1n ∈ U(R) or ann ∈ U(R).
Case 1.1: ann ∈ U(R). Using the elementary row operation R1 − a1na

−1
nnRn → R1, we

have that

A ∼


0

C
...
0

an1 0 · · · 0 ann

 ,



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 22

where

C =


a11 − a1nan1ann

−1 a12 a13 · · · a1,n−1

a21 a22 0 · · · 0
a31 0 a33 · · · 0
...

...
...

. . .
...

an−1,1 0 0 · · · an−1,n−1

 .

Then

det(A) = (−1)n+nann det(C) = ann det(C). (3)

Let

T =




t11 t12 t13 · · · t1,n−1

t21 t22 0 · · · 0
t31 0 t33 · · · 0
...

...
...

. . .
...

tn−1,1 0 0 · · · tn−1,n−1

 ∈ An−1(R)

∣∣∣∣∣∣∣∣∣∣∣
det




t11 − a1nan1a

−1
nn t12 t13 · · · t1,n−1

t21 t22 0 · · · 0
t31 0 t33 · · · 0
...

...
...

. . .
...

tn−1,1 0 0 · · · tn−1,n−1



 = 0


.

Since 
t11 t12 t13 · · · t1,n−1

t21 t22 0 · · · 0
t31 0 t33 · · · 0
...

...
...

. . .
...

tn−1,1 0 0 · · · tn−1,n−1

 ∈ T

if and only if 
t11 − a1nan1a

−1
nn t12 t13 · · · t1,n−1

t21 t22 0 · · · 0
t31 0 t33 · · · 0
...

...
...

. . .
...

tn−1,1 0 0 · · · tn−1,n−1

 ∈ An−1(R, 0),

it follows that |T | = |An−1(R, 0)|. From (3), det(A) = 0 if and only if det(C) = 0. The
number of matrices C with determinant 0 is |T | = |An−1(R, 0)|. The number of choices
for an1 is qe, the number of choices for a1n is qe, and the number of choices for ann is
(q − 1)qe−1. In this case, the possible choices for A is

(q − 1) q3e−1|An−1(R, 0)|.



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 23

Case 1.2: a1n ∈ U(R) and ann /∈ U(R). Let

D =


a11 a12 a13 · · · a1,n−1

a21 a22 0 · · · 0
a31 0 a33 · · · 0
...

...
...

. . .
...

an−1,1 0 0 · · · an−1,n−1

 .

Using the cofactor expansion through the last column of A, it follows that

det(A) = (−1)n+1(−1)n−1+1a1nan1diag(a22, a33, . . . , an−1,n−1) + (−1)n+nann det(D)

= −a1nan1diag(a22, a33, . . . , an−1,n−1) + ann det(D). (4)

It is easily seen that det(A) = 0 whenever an1 = 0 and D ∈ An−1(R, 0). The number of
choices for a1n is (q − 1)qe−1, the number of choices for ann is qe−1, and the number of
choices for D is |An−1(R, 0)|. In this case, the possible choices for A is at least

(q − 1) q2(e−1)|An−1(R, 0)|.

Case 2: ann /∈ U(R) and a1n /∈ U(R). Then the elements in the last column are in γR.
Let B = [bij ] be the matrix in An(R) be defined by

bij =

{
wij if (i, j) ∈ {(1, n), (n, n)}
aij otherwise,

where a1n = γw1n and ann = γwnn for some for some w1n, wnn ∈
e−2∑
j=0

γjV and V is

defined in Lemma 1. Let C = [cij ] be the matrix in An(R/γ
e−1R) defined by cij =

bij + γe−1R. We note that det(A) = γ det(B) ∈ R. Then det(A) = 0 in R if and only if
det(B) ∈ γe−1R which is equivalent to det(C) = 0 + γe−1R in R/γe−1R. For each matrix
C ∈ An(R/γ

e−1R, 0 + γe−1R), there are q3n−4 corresponding matrices B ∈ An(R, 0).
Since the number of possible matrices C is |An(R/γ

e−1R, 0 + γe−1R)| and the matrix A
is uniquely determined by B by multiplying the last column by γ, the number of choices
for A is

q3n−4|An(R/γ
e−1R, 0 + γe−1R)|.

In summary, we have

|An(R, 0)| ≥ (q − 1) q2(e−1)(qe+1 + 1)|An−1(R, 0)|+ q3n−4|An(R/γ
e−1R, 0 + γe−1R)|

as desired. ■
For a FCCR of nilpotency index 2, we have the following bound.



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 24

Corollary 5. Let R be a FCCR of nilpotency index 2 and residue field Fq. If γ is a
generator of the maximal ideal of R, then |A1(R, 0)| = 1 and

|An(R, 0)| ≥ (q − 1)q2(q3 + 1)|An−1(R, 0)|+ q3n−4
(
q3n−2 − q2n−3(q − 1)n(q + (n− 1))

)
for all integers n ≥ 2.

Proof. Clearly, |A1(R, 0)| = 1. Let n ≥ 2 be an integer. We note that R/γe−1R ∼= Fq.
From Proposition 4 and Corollary 2, we have

|An(R, 0)| ≥ (q − 1)q2(q3 + 1)|An−1(R, 0)|+ q3n−4|An(Fq, 0)|
= (q − 1)q2(q3 + 1)|An−1(R, 0)|+ q3n−4

(
q3n−2 − q2n−3(q − 1)n(q + (n− 1))

)
as desired. ■

3.2.2. Singular Arrowhead Matrices over FCCRs with Non-Zero Determinant

In this subsection, an upper bound on the number of n × n singular arrowhead matrices
over R with a fixed non-zero determinant is presented.

First, a relation between |An(R, γ
i)| and |An(R, b)| is derived for all b ∈ γiR \ γi+1R.

Proposition 5. Let R be a FCCR with maximal ideal generated by γ, residue field Fq,
and nilpotency index e. Then

|An(R, γ
i)| = |An(R, b)|

for all b ∈ γiR \ γi+1R and 1 ≤ i < e.

Proof. Let b ∈ γiR \ γi+1R. Then b = aγi for some a ∈ U(R). Let ψ : An(R, γ
i) →

An(R, aγ
i) be the function defined by

ψ(A) = diag(a, 1, 1, . . . , 1)A.

Using the fact that a is convertible and arguments similar to those in the proof of Proposi-
tion 2, it can be deduced that ψ is a bijection from An(R, γ

i) onto An(R, aγ
i). As desired,

|An(R, b)| = |An(R, γ
i)|. ■

Lemma 2. Let R be a FCCR of nilpotency index e ≥ 3 and residue field Fq and let n be
a positive integer. If γ is a generator of the maximal ideal of R, then

|An(R, γ
s)| = q3(n−1)|An(R/γ

e−1R, γs + γe−1R)|

for all 1 ≤ s < e− 1.

Proof. Let 1 ≤ s < e− 1 be an integer and let β : An(R) → An(R/γ
e−1R) be an additive

group homomorphism defined by
β(A) = A,



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 25

where [aij ] := [aij + γe−1R] for all [aij ] ∈ An(R). Note that, for each A ∈ An(R),
det(β(A)) = γs + γe−1R if and only if det(A) = γs + γe−1b for some b ∈ V , where V is
defined in Lemma 1. Since 1 ≤ e − s − 1 < e − 1, it follows that 1 + γe−s−1b is a unit in
U(R). Hence,

|{A ∈ An(R) |det(A) = γs + γe−1b for some b ∈ V }|
= |{A ∈ An(R) | det(A) = γs(1 + γe−s−1b) for some b ∈ V }|
= |{A ∈ An(R) | det(A) = γs}|
= |An(R, γ

s)|.

Equivalently,

|{A ∈ An(R) | det(β(A)) = γs + γe−1R}| = |V ||An(R, γ
s)| = q|An(R, γ

s)|. (5)

Since | ker(β)| = q3n−2, we have

|{A ∈ An(R) | det(β(A)) = γs + γe−1R}|
= | ker(β)||{B ∈ An(R/γ

e−1R) | det(B) = γs + γe−1R}|
= q3n−2|An(R/γ

e−1R, γs + γe−1R)|. (6)

Combining (5) and (6), it can be concluded that

q|An(R, γ
s)| = q3n−2|An(R/γ

e−1R, γs + γe−1R)|.

Therefore,
|An(R, γ

s)| = q3(n−1)|An(R/γ
e−1R, γs + γe−1R)|

as desired. ■
Applying Lemma 2 recursively, the next corollary follows.

Corollary 6. Let R be a FCCR of nilpotency index e+f and residue field Fq, where 2 ≤ e
and 1 ≤ f are integers. If the maximal ideal of R is generated by γ, then

|An(R, γ
s)| = q3f(n−1)|An(R/γ

eR, γs + γeR)|

for all 1 ≤ s < e.

A general recursive formula for the number An(R, γ
s) is presented for all s ≥ 1 in the

next theorem.

Theorem 3. Let R be a FCCR of nilpotency index e and residue field Fq and let n be a
positive integer. If the maximal ideal of R is generated by γ, then

|An(R, γ
s)| = q3(e−s−1)(n−1)

q − 1

(
q3n−2|An(R/γ

sR, 0 + γsR)| − |An(R/γ
s+1R, 0 + γs+1R)|

)
.

for all integers 1 ≤ s < e.



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 26

Proof. Let 1 ≤ s < e be an integer and let µ : An(R/γ
s+1R) → An(R/γ

sR) be an additive
group homomorphism defined by

µ(A) = A,

where [aij + γs+1R] := [aij + γsR] for all [aij + γs+1R] ∈ An(R/γ
s+1R). Then, for each

A ∈ An(R/γ
s+1R), det(µ(A)) = 0 + γsR if and only if det(A) = γsb + γs+1R for some

b ∈ V , where V is defined in Lemma 1. Since | ker(µ)| = q3n−2, we have

q3n−2|An(R/γ
sR, 0 + γsR)| = | ker(µ)||An(R/γ

sR, 0 + γsR)|
= |An(R/γ

s+1R, 0 + γs+1R)|

+
∑

b∈V \{0}

|An(R/γ
s+1R, γsb+ γs+1R)|

= |An(R/γ
s+1R, 0 + γs+1R)|

+ (q − 1)|An(R/γ
s+1R, γs + γs+1R)|

by Proposition 5. Hence, we have

|An(R/γ
s+1R, γs + γs+1R)|

=
1

q − 1

(
q3n−2|An(R/γ

sR, 0 + γsR)| − |An(R/γ
s+1R, 0 + γs+1R)|

)
. (7)

By Corollary 6, we have

|An(R, γ
s)| = |An(R/γ

e+1+(s−e−1)R, γs + γe+1+(s−e−1)R)|
= q3(e−s−1)(n−1)|An(R/γ

s+1R, γs + γs+1R)|. (8)

Combining (7) and (8), we therefore have

|An(R, γ
s)| = q3(e−s−1)(n−1)

q − 1

(
q3n−2|An(R/γ

sR, 0 + γsR)| − |An(R/γ
s+1R, 0 + γs+1R)|

)

as desired. ■
For a FCCR of nilpotency index 2, the following bound on |An(R, a)| is derived for all

a ∈ R \ Fq and positive integers n.

Corollary 7. Let R be a FCCR of nilpotency index 2 and residue field Fq. If the maximal
ideal of R is generated by γ, then |A1(R, a)| = 1 and

|An(R, a)| ≤ (q + 1)q5n−7
(
qn+1 − (q − 1)n(q + (n− 1))

)
− q2(q3 + 1)|An−1(R, 0)|

for all a ∈ R \ Fq and integers n ≥ 2.

Proof. Clearly, |A1(R, a)| = 1. Let n ≥ 2 be an integer. By setting s = 1 in (7), we have

|An(R, a)| = |An(R, γ)|



S. Jitman, P. Modjam / Eur. J. Pure Appl. Math, 17 (1) (2024), 11-29 27

=
1

q − 1

(
q3n−2|An(R/γR, 0 + γR)| − |An(R, 0)|

)
=

1

q − 1

(
q3n−2|An(Fq, 0)| − |An(R, 0)|

)
.

Form the proof of Corollary 5, we have

|An(R, 0)| ≥ (q − 1)q2(q3 + 1)|An−1(R, 0)|+ q3n−4|An(Fq, 0)|

which implies that

|An(R, a)| ≤
1

q − 1

(
q3n−2|An(Fq, 0)| −

(
(q − 1)q2(q3 + 1)|An−1(R, 0)|+ q3n−4|An(Fq, 0)|

))
=

1

q − 1

(
(q3n−2 − q3n−4)|An(Fq, 0)| − (q − 1)q2(q3 + 1)|An−1(R, 0)|

)
=

1

q − 1

(
(q2 − 1)q3n−4|An(Fq, 0)| − (q − 1)q2(q3 + 1)|An−1(R, 0)|

)
= (q + 1)q3n−4|An(Fq, 0)| − q2(q3 + 1)|An−1(R, 0)|.

By Corollary 2, we have

|An(Fq, 0)| = q3n−2 − q2n−3(q − 1)n(q + (n− 1)),

and hence,

|An(R, a)| ≤ (q + 1)q3n−4
(
q3n−2 − q2n−3(q − 1)n(q + (n− 1))

)
− q2(q3 + 1)|An−1(R, 0)|

= (q + 1)q5n−7
(
qn+1 − (q − 1)n(q + (n− 1))

)
− q2(q3 + 1)|An−1(R, 0)|

as desired. ■
We note that, for a FCCR of nilpotency index e = 2, a bound on |An−1(R, 0)| is

determined recursively in Corollary 5.

4. Conclusion and Remarks

The enumeration of arrowhead matrices with prescribed determinant has been estab-
lished over a finite field Fq and a finite commutative chain ring R. Over Fq, the number
of n×n arrowhead matrices with prescribed determinant has been completely determined
for all positive integers n. Subsequently, the number of n×n non-singular arrowhead ma-
trices with prescribed determinant over R has been given for all positive integers n. For
singular arrowhead matrices over R, bounds on the number of n × n singular arrowhead
matrices have been presented. A general set up for an upper bound for the number of
n × n singular arrowhead matrices over R with zero determinant has been given as well
as a lower bound for the number of n × n singular arrowhead matrices over R with a
zero-divisor determinant. For e = 2, rigorous forms of such bounds have been presented.



REFERENCES 28

It would be interesting to derive an explicit formula for the number of n× n singular
arrowhead matrices of a fixed determinant in a FCCR R. In general, the study of n × n
arrowhead matrices with prescribed determinant over more general finite commutative
rings such as principal ideal rings, local rings, and Frobenius rings is another interesting
problem.

Acknowledgements

The authors wold like to thank the anonymous referees for there helpful comments. S.
Jitman was funded by National Research Council of Thailand and Silpakorn University
under Research Grant N42A650381.

References

[1] R P Brent and B D McKay. Determinants and ranks of random matrices over Zm.
Discrete Mathematics, 66:35–49, 1987.

[2] W Cheney and D R Kincaid. Linear Algebra Theory and Applications 2nd edition.
Jones and Bartlett Publishers, 2010.

[3] P Choosuwan, S Jitman, and P Udomkavanich. Determinants of matrices over com-
mutative finite principal ideal rings. Finite Fields and Their Applications, 2017:126–
140, 48.

[4] J. Dubail, T. Botzung, J. Schachenmayer, G. Pupillo, and D. Hagenmüller. Large
random arrowhead matrices: Multifractality, semilocalization, and protected trans-
port in disordered quantum spins coupled to a cavity. Phys. Rev. A, 105:023714, Feb
2022.

[5] B Gilberto and F Flaminio. Finite Commutative Rings and Their Applications.
Springer, 2002.

[6] X Hou. Finite commutative chain rings. Finite Fields and Their Applications, 7:382–
396, 2001.

[7] X Hou, K H Leung, and S L Mab. On the groups of units of finite commutative chain
rings. Finite Fields and Their Applications, 9:20–38, 2003.

[8] S Jitman. Determinants of some special matrices over commutative finite chain rings.
Special Matrices, 8:242–256, 2020.

[9] D A Klain. An intuitive derivation of heron’s formula. The American Mathematical
Monthly, 111:709–712, 2004.

[10] J M Lockhart and W P Wardlaw. Determinants of matrices over the integers modulo
m. Mathematics Magazine, 80:207–214, 2007.



REFERENCES 29

[11] L Lovász. Singular spaces of matrices and their application in combinatorics. Bulletin
Brazilian Mathematical Society, 20:87–99, 1989.

[12] J A Marrero, J N Valdés, and M T Villar. Associating hub-directed multigraphs to
arrowhead matrices. Mathematical Methods in the Applied Sciences, 41(6):2360–2369,
2018.

[13] A Mukhopadhyay. On the probability that the determinant of an n× n matrix over
a finite field vanishes. Discrete Mathematics, 51:311–315, 1984.

[14] H Saberi Najafi, S A Edalatpanah, and G A Gravvanis. An efficient method for
computing the inverse of arrowhead matrices. Applied Mathematics Letters, 33:1–5,
2014.

[15] L Shen and B W Suter. Bounds for eigenvalues of arrowhead matrices and their
applications to hub matrices and wireless communications. EURASIP Journal on
Advances in Signal Processing, 2009(1):379402, 2009.

[16] N J Stor, I Slapničar, and J L Barlow. Accurate eigenvalue decomposition of ar-
rowhead matrices and applications. Linear Algebra and its Applications, 464:62–89,
2015.