Page 26 - programme book

AA-006
On the Properties of the Almost Unimodular Matrix

a) Arif Mandangan 1, a) , Hailiza Kamarulhaili 2,b) , Muhammad Asyraf Asbullah 3,c)

1 Mathematics, Computer Graphics and Visualization Research Laboratory
Faculty of Science and Natural Resources, Universiti Malaysia Sabah
Jalan UMS, 88500 Kota Kinabalu Sabah Malaysia
2 School of Mathematical Sciences,
Universiti Sains Malaysia,
11800 USM Penang, Gelugor, Pulau Pinang, Malaysia.
3 Laboratory of Cryptography, Analysis and Structure,
Institute for Mathematical Sciences, Universiti Putra Malaysia,
43400 UPM Serdang, Serdang, Selangor, Malaysia
3 Centre of Foundation Studies for Agricultural Science, Universiti Putra Malaysia,
43400 UPM Serdang, Serdang, Selangor, Malaysia.

a) arifman@ums.edu.my,
b) hailiza@usm.my,
c) Corresponding author: ma_asyraf@upm.edu.my

Abstract. Matrix inversion is one of the most significant operations on matrix. For any integer matrix
∈ ℤ × , the inverse of this matrix may contain numerous numbers of non-integer entries. These
entries could be long floating-point numbers. Storing, transmitting, or operating such inverse could be
cumbersome especially when the size is large. The only square integer matrix that is guaranteed as
×
having integer matrix as its inverse is unimodular matrix ∈ ℤ with property that det( ) = ±1.
Recently, a new integer matrix that has property almost similar to the unimodular matrix emerges. It is
�
�
known as almost unimodular matrix ∈ ℤ × . Although det� � ≠ ±1, the inverse of this matrix is
proved as containing only a single non-integer entry while the rest are guaranteed as integers. In this
�
paper, some properties of the almost unimodular matrix ∈ ℤ × are discussed. By investigating its
properties, then more of its potential applications could be discovered.
Keywords: Square integer matrix, integer matrix inversion operations, unimodular matrix, almost
unimodular matrix.

24