Ranks of Checkered Pattern Matrices with Applications to Information Embedding and Retrieving

Checkered patterns are characterized by their square structure and the use of only two distinct colors. These colors are typically represented by two types of numerical sets: {1,0} and {1,-1}. Matrices based on {1,0} may seem identical to those based on {1,-1} when forming checkered patterns because the only difference is that the numbers 0 are changed to -1. However, these two kinds of matrices are completely different in a mathematical sense because a matrix using {1,0} has a rank of 2 and a matrix using {1,-1} has a rank of 1. Knowing this difference in advance allows us to reduce the computational effort required for matrix operations such as information embedding and retrieving.