Analysis and Translation of Raffaele Rubini's 1857 'Application of the Theory of Determinants: Note' - Educational Activities Involving Rubini's Article

Salvatore J. Petrilli, Jr. (Adelphi University) and Nicole Smolenski (Adelphi University)

We provide two possible educational activities related to the topics in this article, which could be implemented individually or jointly. These two methods have not been tested; however, we encourage teachers to give them a try with whatever modifications they care to make.

Linear Algebra Class

After learning and proving the basic properties of determinants in a Linear Algebra class, Raffaele Rubini's article, "Application of the Theory of Determinants: Note," could be used to further students' understanding of determinant theory. In particular, his article could show students how to find the determinant of matrices of the form

\[M =\begin{bmatrix}a \pm b & c \pm d\\e \pm f & g \pm h\end{bmatrix}.\]

Most lessons in a typical sophomore-level course do not include content at this level; the matrices that students see have only a single number or variable in each position. One possible lesson could involve the teacher handing out Section 1 from Rubini's article for students to read for homework, possibly together with some numerical examples and preferably following a class discussion of their ideas for how they might approach computing such a determinant. In the subsequent class, students could discuss the article and how Rubini computed a determinant of an \(n\times n\) matrix in this form.

Teachers can utilize the article in this way if an individual student is naturally curious about how he/she could find the determinant of a matrix of such a form or as an extra topic in which teachers utilize the concept from Columbia University's Teachers College of "Stealing Time'' in the mathematics classroom. Stealing time involves "avoiding unnecessary repetition and review'' and replaces that with an advanced topic [Weinberg and Ferrara, 2013, p. 86]. Matrices have shown great potential to fill stolen time [Weinberg and Ferrara, 2013]. Teachers College has demonstrated a variety of success stories of such a nature.

As another example, a teacher could give his/her students equations (10), (11), and (15) from Section 3 of Rubini's article, which give the reader an alternative method to use instead of Laplace expansion when matrices are of a certain form. (Download a sample handout, Rubini and Determinants Classroom Activity.) After students examine the various examples, they can work in groups to determine how they could find a determinant of an \(n\times n\) matrix in such a form and what the benefits are to using the method presented in Rubini's article as opposed to Laplace expansion.

Algebra II or Pre-calculus Class

Rubini's article could also be used as a Mathematics Common Core interdisciplinary activity with high school World History and Algebra II (or Pre-calculus) classes. During the unit in which students learn of the Unification of Italy in World History class, mathematics teachers could have their students do a small project on a restricted list of female and male Italian mathematicians during this historical time period, including Rubini. With female mathematicians such as Maria Gaetana Agnesi (1718–1799), Maria Gramegna (1887–1915), and Pia Nalli (1886–1964) included on the list, students will learn that females also made contributions to the field of mathematics. Excerpts of Mazzotti's [1998] article could be listed as a possible source for students who decide to do their project on mathematicians such as Brioschi, Padula, and Rubini. These students would learn of the schism between synthetic and analytic mathematics in the Italian mathematical community and incorporate it into their brief presentations of their projects.