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How to cite: N. Khasanah and Farikhin, “Counter Example:The Algorithm of Determinant of 

Centrosymmteric Matrix based on Lower Hessenberg Form”, JMM, vol. 6, no. 1, pp. 20-29, May 2020. 

Counter Example: The Algorithm of Determinant of 

Centrosymmteric Matrix based on Lower Hessenberg Form  
 

 

Nur Khasanah1, Farikhin2 

1UIN Walisongo Semarang, khasanah.nur@walisongo.ac.id 
2Universitas Diponegoro, farikhin@undip.ac.id 

 
doi: https://doi.org/10.15642/mantik.2020.6.1.20-29 

 

 
Abstrak: Algoritma dalam perhitungan determinan matriks centrosymetric telah diperoleh 

sebelumnya. Algoritma tersebut menunjukkan tahapan dalam perhitungan komputasi dari determinan 

yang efisien pada matriks centrosymmetric yaitu dengan melakukan perhitungan pada blok 

matriksnya saja. Salah satu blok matriks yamng muncul pada algoritma tersebut adalah matriks 

Hessenberg bawah. Meskipun, bentuk matriks lainnya juga memungkinkan muncul pada perhitungan 

determianan  matriks centrosymmetric. Oleh sebab itu, artikel ini bertujuan untuk menunjukan 

kemungkinan kemunculan blok matriks centrosymmetric  dan bagaimana algoritma yang telah 

diperoleh, diterapkan dalam menyelesaikan determinan berbagai jenis matriks centrosymmetric. 

Beberapa contoh dari blok matriks yang berbeda pada determinanan matriks centrosymmetric 

diberikan pula.  Contoh tersebut sangat bermanfaat dalam pemahaman lanjut saat menggunakan 

algoritma ini dengan berbagai kasus yang berbeda. 

 

Kata kunci: Centrosymmetric; Determinan; Matriks blok; Hessenberg bawah 

 

 
Abstract: The algorithm for computing determinant of centrosymmetric matrix has been evaluated 

before. This algorithm shows the efficient computational determinant process on centrosymmetric 

matrix by working on block matrix only. One of block matrix at centrosymmetric matrix appearing 

on this algorithm is lower Hessenberg form. However, the other block matrices may possibly appear 

as block matrix for centrosymmetric matrix’s determinant. Therefore, this study is aimed to show the 

possible block matrices at centrosymmetric matrix and how the algorithm solve the centrosymmetric 

matrix’s determinant. Some numerical examples for different cases of block matrices on determinant 

of centrosymmetric matrix are given also. These examples are useful for more understanding for 

applying the algorithm with different cases. 
 

Keywords: Centrosymmetric; Determinant; Block matrices; Lower Hessenberg 

 

 

 

  

  

Jurnal Matematika MANTIK 

Volume 6, No. 1, May 2020, pp. 20-29 

 

ISSN: 2527-3159 (print) 2527-3167 (online) 

http://u.lipi.go.id/1458103791
http://u.lipi.go.id/1457054096


N. Khasanah and Farikhin 

Counter Example:The Algorithm of Determinant of Centrosymmteric Matrix based on Lower Hessenberg 

Form 

21 

1. Introduction 

Centrosymmetric matrix is the matrix with a special structure at its entries that is 

symmetry about its center. Some special properties of centrosymmetric matrix about its 

special structure are observed before [1-9]. Based on the role of orthogonal matrix and 

special structure of centrosymmetric matrix, the centrosymmetric matrix can be formed as 

block matrices. These block matrices can be used to construct an efficient algorithm on 

determinant. The study at [10-14] shows the computational process for inverse and 

determinant at special matrix, called centrosymmetric matrix. 

Moreover, [15] also working at the analytical process for determinant of 

centrosymmetric matrix with lower Hessenberg as block matrix. The result of the study is 

only computing block matrix at computational process for determinant centrosymmetric 

matrix having a half size of centrosymmetric’s size. However, this analytical process only 

aplicable at one example with lower Hesssenberg at certain block matrix at 

centrosymmetric matrix. 

 On the other side, based on its applications, the other block matrices also appear at 

centrosymmetric matrix. By the roles of centrosymmetric matrix [16], then the evaluation 

of previous algorithm at some block matrices is needed.  The different block matrices that 

possibly appear are upper Hessenberg, centrosymmetric, lower triangular, upper triangular, 

diagonal, tridiagonal matrix also appears on its. Based on previous study, the algorithm of 

determinant centrosymmetric matrix can not be applied at general centrosymmetric matrix 

which will be evaluated at this paper. This study shows the possible different block matrices 

appear at this matrix and the different treatment for applying the algorithm of determinant 

of centrosymmetric matrix. 

 

2. Preliminaries 

In this section, there are some basic properties of centrosymmetric matrix before 

further discussion on determinant of centrosymmetric matrix. 

Definition 1 [14,15]. The 𝐴 = (𝑎𝑖𝑗)𝑛×𝑛
∈ 𝑅𝑛×𝑛 is a centrosymmetric matrix, if  

 𝑎𝑖𝑗 = 𝑎𝑛−𝑖+1,𝑛−𝑗+1, 1 ≤ 𝑖 ≤ 𝑛,1 ≤ 𝑗 ≤ 𝑛 or equivalently 𝑱𝒏𝑨𝑱𝒏 = 𝑨, (1) 

with 𝑱𝒏 = (𝒆𝒏,𝒆𝒏−𝟏,⋯,𝒆𝟏) and 𝒆𝒊 is the unit vector with the i-th elements 1 and others 0. 
Furthermore, based on the definition of centrosymmetric matrix, by using orthogonal 

matrix, the properties of this matrix is written as : 

Lemma 2 [14,15]. The 𝑨 = (𝑎𝑖𝑗)𝑛×𝑛
∈ 𝑅𝑛×𝑛(𝑛 = 2𝑚) is centrosymmetric matrix, if and 

only if 𝑨 has the form: 
 

 𝐴 = (
𝐵 𝐽𝑚𝐶𝐽𝑚
𝐶 𝐽𝑚𝐵𝐽𝑚

), and 𝑄𝑇𝐴𝑄 = (
𝐵 −𝐽𝑚𝐶 0𝑚
0𝑚 𝐵 +𝐽𝑚𝐶

)   (2)  

 

where 𝑩 ∈ 𝑅𝑚×𝑚, 𝑪 ∈ 𝑅𝑚×𝑚 and 𝑄 =
√2

2
(
𝐼𝑚 𝐼𝑚
−𝐽𝑚 𝐽𝑚

). 

 

Proof. 𝑄𝑇𝐴𝑄 =
√2

2
(
𝐼𝑚 −𝐽𝑚
𝐼𝑚 𝐽𝑚

)(
𝐵 𝐽𝑚𝐶𝐽𝑚
𝐶 𝐽𝑚𝐵𝐽𝑚

)
√2

2
(
𝐼𝑚 𝐼𝑚
−𝐽𝑚 𝐽𝑚

) = (
𝐵 − 𝐽𝑚𝐶 0𝑚
0𝑚 𝐵 +𝐽𝑚𝐶

).∎ 

 

Therefore, the computing determinant block matrix only, 𝑩−𝑱𝒎𝑪 and 𝑩+ 𝑱𝒎𝑪, is 
same as computing determinant of centrosymmetric matrix. 

 

  



Jurnal Matematika MANTIK 

Volume 6, Issue 1, May 2020, pp. 20-29 
 

22 

3. Results and Discussion 

3.1 The Algorithm of Determinant of Centrosymmetric Matrix 

Based on the algorithm of determinant of lower Hessenberg matrix, this algorithm can 

be used to compute the determinant of centrosymmetric matrix with lower Hessenberg as 

block matrices. The steps of determinant of centrosymmetric matrix with lower Hessenberg 

form as block matrices are written as follows [14,15]. 

a. Construct block centrosymmetric matrix 
By the Lemma 2 and based on orthogonal matrix then the centrosymmetric matrix is 

written as block centrosymmetric matrix (
𝑩−𝑱𝒎𝑪 𝟎𝒎
𝟎𝒎 𝑩+ 𝑱𝒎𝑪

). It can be seen that 

centrosymmetric matrix has block matrices 𝑩−𝑱𝒎𝑪 and 𝑩+𝑱𝒎𝑪. 
b. Construct block centrosymmetric matrix on Hessenberg matrix 

By the previous explanation, centrosymmetric has block matrices 𝑩− 𝑱𝒎𝑪 and 𝑩+
𝑱𝒎𝑪. These matrices have special form as lower Hessenberg matrix. Based on the 
definition of centrosymmetric matrix and lower Hessenberg matrix, then the 

centrosymmetric matrix with lower Hessenberg as block matrices has the form as 

follows. 

𝑨 = (
𝑩 𝑱𝒎𝑪𝑱𝒎
𝑪 𝑱𝒎𝑩𝑱𝒎

) 

 







































=

−−−

−−−−−−

−−

−−

−−−−−−

−−−

11122,1,

2122233,12,11,1

1,12,11,1,1,21,22221

1,2,1,,,11,11211

11121,1,1,1,2,1,

21221,2,2,11,12,11,1

1,12,13,1232221

1,2,1211

bbcc

bbbccc

bbbbcccc

bbbbcccc

ccccbbbb

ccccbbbb

cccbbb

ccbb

mm

mmm

mmmmmmmm

mmmmmmmm

mmmmmmmm

mmmmmmmm

mmm

mm













 

Then, the block matrices of this matrix are 

𝑩 =

(

  
 

𝑏11 𝑏12
𝑏21 𝑏22 𝑏23
⋮ ⋮ ⋱ ⋱

𝑏𝑚−1,1 𝑏𝑚−1,2 ⋯ 𝑏𝑚−1,𝑚−1 𝑏𝑚−1,𝑚
𝑏𝑚,1 𝑏𝑚,2 ⋯ 𝑏𝑚,𝑚−1 𝑏𝑚,𝑚 )

  
 

 and 

 𝑱𝒎𝑪 =

(

 
 

𝑐11 𝑐12
𝑐21 𝑐22 𝑐23
⋮ ⋮ ⋱ ⋱

𝑐𝑚−1,1 𝑐𝑚−1,2 ⋯ 𝑐𝑚−1,𝑚−1 𝑐𝑚−1,𝑚
𝑐𝑚,1 𝑐𝑚,2 ⋯ 𝑐𝑚,𝑚−1 𝑐𝑚,𝑚 )

 
 

  

 



N. Khasanah and Farikhin 

Counter Example:The Algorithm of Determinant of Centrosymmteric Matrix based on Lower Hessenberg 

Form 

23 

where both of them are Hessenberg matrices. There is special form as block matrices 

on centrosymmetric matrix. This condition is useful on computing determinant of 

centrosymmetric matrix using determinant of lower Hessenberg matrix. 

From Lemma 2, there is an orthogonal matrix 𝑸 =
√2

2
(
𝑰𝒎 𝑰𝒎
−𝑱𝒎 𝑱𝒎

), then 

 𝑸𝑻𝑨𝑸 = (
𝑩−𝑱𝒎𝑪 𝟎𝒎
𝟎𝒎 𝑩+ 𝑱𝒎𝑪

) = (
𝑴 𝟎𝒎
𝟎𝒎 𝑵

)    (3) 

where 𝑴 = 𝑩−𝑱𝒎𝑪, 𝑵 = 𝑩+ 𝑱𝒎𝑪 and 𝑴,𝑵 are Hessenberg matrices. 
By the same way [16] about the determinant of lower Hessenberg matrix, then we can 

assume 

 �̃�−𝟏 = (
𝜶𝑴 𝑾𝑴
𝑟𝑀 𝜷𝑴

𝛵 ), �̃�
−𝟏 = (

𝜶𝑵 𝑾𝑵
𝑟𝑁 𝜷𝑵

𝛵 )    (4) 

and  

 �̃� = (
𝒆𝟏
𝛵 0
𝑴 𝒆𝒎

), �̃� = (
𝒆𝟏
𝛵 0
𝑵 𝒆𝒎

).     (5) 

Therefore 𝑟𝑀,𝑟𝑁 ≠ 0, if 𝑟𝑀, 𝑟𝑁 = 0 than 𝜶𝑴,𝜶𝑵 = 0 since 𝑴𝜶𝑴 = 0 and 𝑵𝜶𝑵 =
0. It implies that �̃�−𝟏 and �̃�−𝟏 are nonsingular matrix, which are contradiction. 

c. Compute determinant of centrosymmetric matrix 
The block of centrosymmetric matrix as explained before, it is formed as 

 𝑨 = 𝑸(
𝑩−𝑱𝒎𝑪 𝟎𝒎
𝟎𝒎 𝑩+𝑱𝒎𝑪

)𝑸𝛵 = 𝑸(
𝑴 𝟎𝒎
𝟎𝒎 𝑵

)𝑸𝛵   (6) 

then  

 𝑑𝑒𝑡(𝑨) = 𝑑𝑒𝑡(𝑸) ⋅𝑑𝑒𝑡(
𝑴 𝟎𝒎
𝟎𝒎 𝑵

)⋅𝑑𝑒𝑡(𝑸𝛵) = 𝑑𝑒𝑡(𝑴) ⋅𝑑𝑒𝑡(𝑵). (7) 

The theorem of determinant of centrosymmetric matrix with lower Hessenberg as 

block matrices is obtained as follows. 

Theorem 3 [14,15]. Let 𝑨 is centrosymmetric matrix and it’s block matrices are 𝑴,𝑵as 
Hessenberg matrices which are described before, then 

𝑑𝑒𝑡(𝑨) = 𝑟𝑁 ⋅ 𝑟𝑀 ⋅∏ (𝑔𝑖,𝑖+1 ⋅ 𝑞𝑖,𝑖+1)
𝑚−1
𝑖=1 .    (8) 

This step is the algorithm determinant of centrosymmetric matrix with lower 

Hessenberg as block matrix.  The other blocks matrices also appear on centrosymmetric 

matrix are upper Hessenberg, centrosymmetric, lower triangular, upper triangular, 

diagonal, tridiagonal matrix. Furthermore, some numerical examples of determinant of 

centrosymmetric matrix with different block matrices are given. 

 

3.2 Numerical Experiences 

This part will show the different block matrices arising at some examples of 

centrosymmetric matrices by applying the previous algorithm. 

Example 1. [14,15] Given the following centrosymmetric matrix  

𝑨 =

(

 
 
 
 
 

1 1 0 0 0 0 2 1
0 2 2 0 0 1 3 0
1 2 2 1 2 2 2 1
1 0 1 1 4 1 3 2
2 3 1 4 1 1 0 1
1 2 2 2 1 2 2 1
0 3 1 0 0 2 2 0
1 2 0 0 0 0 1 1)

 
 
 
 
 

where 𝑩 = (

1 1 0 0
0 2 2 0
1 2 2 1
1 0 1 1

)is lower Hessenberg as  



Jurnal Matematika MANTIK 

Volume 6, Issue 1, May 2020, pp. 20-29 
 

24 

block matrix, 𝑪 = (

2 3 1 4
1 2 2 2
0 3 1 0
1 2 0 0

) and 𝑱𝟒 = (

0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0

).  

Therefore, 𝑴 = 𝑩−𝑱𝟒𝑪 = (

0 −1 0 0
0 −1 1 0
0 0 0 −1
−1 −3 0 −3

), 𝑵 = 𝑩+𝑱𝟒𝑪 = (

2 3 0 0
0 5 3 0
2 4 4 3
3 3 2 5

) are 

lower Hessenberg matrices.  

Based on the algorithm before, this step can be continued caused by both of block matrices 

are lower Hessenberg form. 

Determinant of this matrix can be solved with �̃� =

(

 
 

1 0 0 0 0
0 −1 0 0 0
0 −1 1 0 0
0 0 0 −1 0
−1 −3 0 −3 1)

 
 

,  

�̃� =

(

 
 

1 0 0 0 0
2 3 0 0 0
0 5 3 0 0
2 4 4 3 0
3 3 2 5 1)

 
 

. So, it can be founded �̃�−𝟏 =

(

 
 

1 0 0 0 0
0 −1 0 0 0
0 −1 1 0 0
0 0 0 −1 0
−1 −3 0 −3 1)

 
 

 and  

�̃�−𝟏 =

(

 
 

1 0 0 0 0
−0.6667 0.3333 0 0 0
1.1111 −0.5556 0.3333 0 0
−1.2593 0.2963 −0.4444 0.3333 0
3.0741 −1.3704 1.5556 −1.6667 1)

 
 

.  

Therefore, 

𝑟𝑀 = 1,𝑟𝑁 = 3.0741 and ∏ (𝑔𝑖,𝑖+1)
4
𝑖=1 = (−1)(1)(−1) = 1, ∏ (𝑞𝑖,𝑖+1)

4
𝑖=1 =

(3)(3)(3) = 27 

then  

𝑑𝑒𝑡(𝑯) = 𝑟𝑀 ⋅ 𝑟𝑁 ⋅ ∏(𝑔𝑖,𝑖+1 ⋅ 𝑞𝑖,𝑖+1)

𝑚−1

𝑖=1

= (1)(3.0741)(1)(27) = 83.0007. 

Example 2. Consider the centrosymmetric matrix 𝑨 =

(

 
 
 
 
 

2 3 1 4 1 1 0 1
1 2 2 2 1 2 2 1
0 3 1 0 0 2 2 0
1 2 0 0 0 0 1 1
1 1 0 0 0 0 2 1
0 2 2 0 0 1 3 0
1 2 2 1 2 2 2 1
1 0 1 1 4 1 3 2)

 
 
 
 
 

  

with 𝑩 is not lower Hessenberg as block matrix. By Lemma 2, the matrix of 𝑨 is formed 
as block matrices  

𝑩−𝑱𝟒𝑪 = 𝑴 = (

1 3 0 3
0 0 0 1
0 1 −1 0
0 1 0 0

), 𝑩+𝑱𝟒𝑪 = 𝑵 = (

3 3 2 5
2 4 4 3
0 5 3 0
2 3 0 0

)  



N. Khasanah and Farikhin 

Counter Example:The Algorithm of Determinant of Centrosymmteric Matrix based on Lower Hessenberg 

Form 

25 

and we have 

�̃� =

(

 
 

1 0 0 0 0
1 3 0 3 0
0 0 0 1 0
0 1 −1 0 0
0 1 0 0 1)

 
 

, �̃� =

(

 
 

1 0 0 0 0
3 3 2 5 0
2 4 4 3 0
0 5 3 0 0
2 3 0 0 1)

 
 

  

which are not lower triangular matrices form, then the algorithm is not suitable for this 

condition. This algorithm is stopped because of non lower Hessenberg of its block matrices 

𝑴 and 𝑵. 

Example 3. Consider centrosymmetric matrix 𝑨 =

(

 
 
 
 
 

1 1 0 0 4 1 3 2
0 2 2 0 2 2 2 1
1 2 2 1 0 1 3 0
1 0 1 1 0 0 2 1
1 2 0 0 1 1 0 1
0 3 1 0 1 2 2 1
1 2 2 2 0 2 2 0
2 3 1 4 0 0 1 1)

 
 
 
 
 

 has 𝑩  

and 𝑪 are lower Hessenberg as block matrices. Thus, we have 

𝑴 = (

−1 −2 −1 −4
−1 0 0 −2
1 −1 1 1
0 −2 1 1

), 𝑵 = (

3 4 1 4
1 4 4 2
1 5 3 1
2 2 1 1

), then we have  

�̃� =

(

 
 

1 0 0 0 0
−1 −2 −1 −4 0
−1 0 0 −2 0
1 −1 1 1 0
0 −2 1 1 1)

 
 

, �̃� =

(

 
 

1 0 0 0 0
3 4 1 4 0
1 4 4 2 0
1 5 3 1 0
2 2 1 1 1)

 
 

  

which are not lower triangular matrices form, then the algorithm is not suitable for this 

condition caused by the lower Hessenberg matrices appear on two block matrices 𝑩, 𝑪. 

Example 4. Given 𝑨 =

(

 
 
 
 
 

2.5 1.5 0 0 2.5 1.5 1 0
1.5 2.5 1.5 0 1.5 2.5 1.5 1
1 1.5 2.5 1.5 1 1.5 2.5 1.5
0 1 1.5 2.5 0 1 1.5 2.5
2.5 1.5 0 0 2.5 1.5 1 0
1.5 2.5 1.5 0 1.5 2.5 1.5 1
1 1.5 2.5 1.5 1 1.5 2.5 1.5
0 1 1.5 2.5 0 1 1.5 2.5)

 
 
 
 
 

 then   

𝑴 = 𝑩−𝑱𝟒𝑪 = (

2.5 0.5 −0.5 −2.5
0.5 1 −1 −0.5
−0.5 −1 1 0.5
−2.5 −0.5 0.5 2.5

),  

𝑵 = 𝑩+ 𝑱𝟒𝑪 = (

2.5 2.5 2.5 2.5
2.5 4 4 2.5
2.5 4 4 2.5
2.5 2.5 2.5 2.5

) are centrosymmetric matrices too. 

It can be seen that centrosymmetric matrix can appear on centrosymmetric matrix as block 

matrix. It seen that block matrices of matrix are 𝑩 = 𝑪 cause be zero on determinant and 
the algorithm is not suitable for this condition. 



Jurnal Matematika MANTIK 

Volume 6, Issue 1, May 2020, pp. 20-29 
 

26 

Example 5. Given 𝑨 =

(

 
 
 
 
 

1 2 3 4 1 0 4 3
2 1 2 3 0 1 0 4
3 2 1 2 4 0 1 0
4 3 2 1 3 4 0 1
1 0 4 3 1 2 3 4
0 1 0 4 2 1 2 3
4 0 1 0 3 2 1 2
3 4 0 1 4 3 2 1)

 
 
 
 
 

, then  

𝑴 = 𝑩−𝑱𝟒𝑪 = (

−2 −2 3 3
−2 1 1 3
3 1 1 −2
3 3 −2 −2

), 𝑵 = 𝑩+ 𝑱𝟒𝑪 = (

4 6 3 5
6 1 3 3
3 3 1 6
5 3 6 4

) are 

centrosymmetric matrices too. 

It can be seen that centrosymmetric matrix can appear on centrosymmetric matrix as block 

matrix. It can be seen that block matrices of matrix are 𝑩 ≠ 𝑪 caused determinant is 
nonzero and the algorithm is not suitable for this condition. Moreover, there no lower 

Hessenberg form as block matrix at matrix 𝑩. 

Example 6. Consider the centrosymmetric matrix 𝑨 =

(

 
 
 
 
 

1 0 0 0 3 1 3 2
2 3 0 0 0 3 2 2
3 2 1 0 0 0 2 1
2 3 1 2 0 0 0 2
2 0 0 0 2 1 3 2
1 2 0 0 0 1 2 3
2 2 3 0 0 0 3 2
2 3 1 3 0 0 0 1)

 
 
 
 
 

 

has tridiagonal matrix 𝑩 and 𝑪 as block matrices. By Lemma 2, the matrix of 𝑨 is formed 
as block matrices  

𝑩−𝑱𝟒𝑪 = 𝑴 = (

−1 −3 −1 −3
0 1 −3 0
2 0 1 0
0 3 1 2

), 𝑩+𝑱𝟒𝑪 = 𝑵 = (

3 3 1 3
4 5 3 0
4 4 1 0
4 3 1 2

)  

and we have 

�̃� =

(

 
 

1 0 0 0 0
−1 −3 −1 −3 0
0 1 −3 0 0
2 0 1 0 0
0 3 1 2 1)

 
 

, �̃� =

(

 
 

1 0 0 0 0
3 3 1 3 0
4 5 3 0 0
4 4 1 0 0
4 3 1 2 1)

 
 

  

which are not lower triangular matrices form, then the algorithm is not suitable for this 

condition. The algorithm cannot be continued caused the block matrix is not lower 

Hessenberg form. 

Example 7. Consider the centrosymmetric matrix  

𝑨 =

(

 
 
 
 
 

1 2 3 2 3 0 0 0
0 3 2 3 1 3 0 0
0 0 1 1 3 2 2 0
0 0 0 2 2 2 1 2
2 1 2 2 2 0 0 0
0 2 2 3 1 1 0 0
0 0 3 1 3 2 3 0
0 0 0 3 2 3 2 1)

 
 
 
 
 

 where block matrices 𝑩 and 𝑪 are upper Hessenberg 



N. Khasanah and Farikhin 

Counter Example:The Algorithm of Determinant of Centrosymmteric Matrix based on Lower Hessenberg 

Form 

27 

form. By Lemma 2, the matrix of 𝑨 is formed as block matrices  

𝑩−𝑱𝟒𝑪 = 𝑴 = (

1 2 3 −1
0 3 −1 2
0 −2 −1 −2
−2 −1 −2 0

), 𝑩+𝑱𝟒𝑪 = 𝑵 = (

1 2 3 5
0 3 5 4
0 2 3 4
2 1 2 4

)  

and we have 

�̃� =

(

 
 

1 0 0 0 0
1 2 3 −1 0
0 3 −1 2 0
0 −2 −1 −2 0
−2 −1 −2 0 1)

 
 

, �̃� =

(

 
 

1 0 0 0 0
1 2 3 5 0
0 3 5 4 0
0 2 3 4 0
2 1 2 4 1)

 
 

.  

which are not lower triangular matrices form, then the algorithm is not suitable for this 

condition. It happen caused by block matrix 𝑩 is not lower Hessenberg matrix. 

Example 8. Consider the centrosymmetric matrix  

𝑨 =

(

 
 
 
 
 

1 0 0 0 8 0 0 0
0 2 0 0 0 7 0 0
0 0 3 0 0 0 6 0
0 0 0 4 0 0 0 5
5 0 0 0 4 0 0 0
0 6 0 0 0 3 0 0
0 0 7 0 0 0 2 0
0 0 0 8 0 0 0 1)

 
 
 
 
 

where 𝑩 and 𝑪 are diagonal matrices as block matrix.  

By Lemma 2, the matrix of 𝑨 is formed as block matrices  

𝑩−𝑱𝟒𝑪 = 𝑴 = (

1 0 0 −8
0 2 −7 0
0 −6 3 0
−5 0 0 4

), 𝑩+𝑱𝟒𝑪 = 𝑵 = (

1 0 0 8
0 2 7 0
0 6 3 0
5 0 0 4

)  

and we have 

�̃� =

(

 
 

1 0 0 0 0
1 0 0 −8 0
0 2 −7 0 0
0 −6 3 0 0
−5 0 0 4 1)

 
 

, �̃� =

(

 
 

1 0 0 0 0
1 0 0 8 0
0 2 7 0 0
0 6 3 0 0
5 0 0 4 1)

 
 

.  

which are not lower triangular matrices form, then the algorithm is not suitable for this 

condition. It cannot be continued because of lower Hessenberg form does not appear as 

block matrix. 

Example 9. Consider the centrosymmetric matrix 𝑨 =

(

 
 
 
 
 

2 1 0 0 4 3 0 0
1 2 1 0 3 4 3 0
0 1 2 1 0 3 4 3
0 0 1 2 0 0 3 4
4 3 0 0 2 1 0 0
3 4 3 0 1 2 1 0
0 3 4 3 0 1 2 1
0 0 3 4 0 0 1 2)

 
 
 
 
 

  

has 𝑩 and 𝑪are tridiagonal matrix as block matrices. By Lemma 2, the matrix of 𝑨 is formed 
as block matrices  



Jurnal Matematika MANTIK 

Volume 6, Issue 1, May 2020, pp. 20-29 
 

28 

𝑩−𝑱𝟒𝑪 = 𝑴 = (

2 1 −3 −4
1 −1 −3 −3
−3 −3 −1 1
−4 −3 1 2

), 𝑩+𝑱𝟒𝑪 = 𝑵 = (

2 1 3 4
1 5 5 3
3 5 5 1
4 3 1 2

)  

and we have 

�̃� =

(

 
 

1 0 0 0 0
2 1 −3 −4 0
1 −1 −3 −3 0
−3 −3 −1 1 0
−4 −3 1 2 1)

 
 

, �̃� =

(

 
 

1 0 0 0 0
2 1 3 4 0
1 5 5 3 0
3 5 5 1 0
4 3 1 2 1)

 
 

.  

which are not lower triangular matrices form, then the algorithm is not suitable for this 

condition. 

 

4. Conclusions 

The algorithm of determinant centrosymmetric matrix is presented efficiently. This 

algorithm focusing with the centrosymmetric matrix with lower Hessenberg matrix as 

block matrix. But there are some block matrices appear as block of centrosymmetric 

matrices such as upper Hessenberg, centrosymmetric, lower triangular, upper triangular, 

diagonal, tridiagonal matrix. Therefore, this algorithm is only on compute the determinant 

of centrosymmetric matrix with lower Hessenberg as block matrix. For the other block 

matrices, this algorithm does not work well. 

 

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29 

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