In educational research, a fundamental goal is identifying which skills students have mastered, which skills they have not, and which skills they are in the process of mastering. As the number of examinees, items, and skills increases, the estimation of even simple cognitive diagnosis models becomes difficult. We adopt a faster, simpler approach:…

In this note, a method of converting a rhotrix to a special form of matrix termed a "coupled matrix" is proposed. The special matrix can be used to solve various problems involving n x n and (n - 1) x (n - 1) matrices simultaneously.

The Hill cipher is the simplest example of a "block cipher," which takes a block of plaintext as input, and returns a block of ciphertext as output. Although it is insecure by modern standards, its simplicity means that it is well suited for the teaching of such concepts as encryption modes, and properties of cryptographic hash functions. Although…

Most model fit analyses in cognitive diagnosis assume that a Q matrix is correct after it has been constructed, without verifying its appropriateness. Consequently, any model misfit attributable to the Q matrix cannot be addressed and remedied. To address this concern, this paper proposes an empirically based method of validating a Q matrix used…

Using the elementary tools of matrix theory, we show that the product of two rotations in the three-dimensional Euclidean space is a rotation again. For this purpose, three types of rotation matrices are identified which are of simple structure. One of them is the identity matrix, and each of the other two types can be uniquely characterized by…

If M[subscript ij] are minors of an n x n determinant D with elements a[subscript ij] (1 less than or equal to i,j less than or equal to n), then we prove the following relationship [vertical bar]M[subscript k][vertical bar] = [vertical bar]D[vertical bar][superscript k-1]delta[subscript k] where M[subscript k] is any square sub matrix of order k…

In this article, the authors have 2 aims. First, hierarchical, nonhierarchical, and nonstandard log-linear models are defined. Second, application scenarios are presented for nonhierarchical and nonstandard models, with illustrations of where these scenarios can occur. Parameters can be interpreted in regard to their formal meaning and in regard…

This paper presents the row-column multiplication of rhotrices that are of high dimension. This is an extension of the same multiplication carried out on rhotrices of dimension three, considered to be the base rhotrices.

The mathematics for finding and plotting the locations of stars and constellations are available in many books on astronomy, but the steps involve mystifying and fragmented equations, calculations, and terminology. This paper will introduce an entirely new unified and cohesive technique that is easy to understand by mathematicians, and simple…

Model evaluation in covariance structure analysis is critical before the results can be trusted. Due to finite sample sizes and unknown distributions of real data, existing conclusions regarding a particular statistic may not be applicable in practice. The bootstrap procedure automatically takes care of the unknown distribution and, for a given…

This article shows how a matrix can be used to represent a food chain and how the square of this matrix represents the indirect food sources for each animal in the chain. By exploring, through mathematics, the implications when the bottom of the food chain is destroyed, students will see an important connection between mathematics and science.…

As is known, a semi-magic square is an "n x n" matrix having the sum of entries in each row and each column equal to a constant. This note generalizes this notion and introduce a special class of block matrices called "block magic rectangles." It is proved that the Moore-Penrose inverse of a block magic rectangle is also a block magic rectangle.

In this paper "Derive" functions are provided for the computation of the Moore-Penrose inverse of a matrix, as well as for solving systems of linear equations by means of the Moore-Penrose inverse. Making it possible to compute the Moore-Penrose inverse easily with one of the most commonly used Computer Algebra Systems--and to have the blueprint…

This note presents geometric and physical interpretations of the sufficient condition for a critical point to be a strict relative extremum: f[subscript xx]f[subscript yy] - f[superscript 2][subscript xy] greater than 0. The role of the double derivative f[subscript xy] in this inequality will be highlighted in these interpretations. (Contains 14…

A simple mathematical model for the behaviour of how vehicles follow each other along a looped stretch of road is described. The resulting coupled first order differential equations are solved using appropriate matrix techniques and the physical significance of the model is discussed. A number possible classroom exercises are suggested to help…