Determine the Determinant of 4xn Non-Square Matrix Using Radić’s Determinant

Abstract

A non-square matrix is a matrix that has a different number of rows and columns. In the modified double-guard Hill cipher algorithm, a non-square matrix is used as the private key matrix that plays a role in the message encryption and decryption process. Therefore, the determinant of the key matrix is needed to obtain the inverse of the key matrix. Mirko Radić defined the determinant of matrix Amxn, m 4, using Radić’s determinant and an example of its use. The result of this research are the following theorem. If a non-square matrix R = [\matrix (1&0&0&...&0&0@0&1&0&...&0&0@0&a_1&a_2&...&a_i&0@0&0&0&...&0&1)], ai ∈ R, ∀i=1,2,...,n-2 where n>4 then |R|= Σ (-1)i+1 ai, for n odd and Σ (-1)i ai, for n even where i=2 to n-2. The use of the theorem is shown in an example problem using the modified double-guard Hill cipher where matrix R is chosen as the private key matrix. Several conditions must be met by the matrix R to be selected as the key matrix, including all elements of matrix R being positive integers, |R|\neq 0, and R invertible in modulo 128.

Keywords

• Determinant
• Non-Square Matrix
• Radić’s Determinant
• Modified Double-Guard Hill Cipher