Analysis and Translation of Raffaele Rubini's 1857 'Application of the Theory of Determinants: Note' - Formulas and Relations

After delivering a short introductory paragraph, Rubini delved right into formulas for special determinants and relations between them. Intriguingly, he did this without providing the reader with any of the basic properties of determinants or any explanation of his notation. In order for his readers to understand this "Note", it seems as if Rubini assumed they would be knowledgeable about determinants or have access to Brioschi's textbook, La teorica dei determinanti e le sue applicazioni, published in 1854, which was the first Italian work about determinants available to mathematicians in the Kingdom of Two Sicilies. Rubini opened his first section with a reference to this influential work and provided his readers with formulas which are given on page 44 of Brioschi's work [1854]. Although the modern-day reader may be perplexed upon viewing the first determinant Rubini supplied, an individual with knowledge of expansion by minors and properties of determinants would be able to understand the majority of the mathematics presented.  

After reading a couple of sections of Rubini's article, the reader will quickly discover that the author established few connections between the formulas he provided. Most ideas do not lead to the successive idea. Furthermore, where connections could be made within the paper, Rubini chose to format his article in such a way that these ideas were presented disjointly. For instance, in Section 3 of his article, Rubini supplied formulas (8) and (9), respectively:

\[{\begin{vmatrix}1 & 1 & \ldots & 1\\ 1 & 1+x & \ldots & 1\\\vdots & \vdots & \ddots & \vdots\\1 & 1 & \ldots & 1+x \end{vmatrix}}_{n-1}= x^{n-1}\quad\quad\quad(8)\]

and

\[{\begin{vmatrix}1+x & 1 & \ldots & 1\\1 & 1+x & \ldots & 1\\\vdots & \vdots & \ddots & \vdots\\1 & 1 & \ldots & 1+x \end{vmatrix}} _{n} = nx^{n-1} + x^{n},\quad\quad(9)\]

with the indices of \(n\) and \(n-1\) indicating the number of \(x\)'s along the main diagonal, but with both matrices of the \(n\)th order. Formulas (8) and (9) are very similar; the difference is that the matrix of formula (9) has \(x\)'s in every position on the main diagonal, while the matrix of formula (8) has its first row and column composed only of ones. Rubini's notation may seem confusing at first, but after comparing the matrices of formulas (8) and (9), it can be understood that these matrices are of the same order. As he provided no concrete examples in his article, we provide such an example here. Using \(2\times 2\) matrices with \(x = 3\) and computing these determinants by the Laplace expansion, the first formula can be illustrated by the following example:

\[{\begin{vmatrix}1 & 1\\1 & 4 \end{vmatrix}}_{1} = (1)(4) - (1)(1) = 3= 3^1,\]

and the second formula by the example below:

\[{\begin{vmatrix}4 & 1\\1 & 4 \end{vmatrix}}_{2} = (4)(4) - (1)(1) = 15= 2\cdot{3^1} + 3^2.\]

Note that, according to the first formula, for a \(2\times 2\) matrix, the determinant will always be equal to the value of \(x.\)

Now, substituting \(x\) for \(1 + x\) and \(x - 1\) for \(x\) in formulas (8) and (9), we obtain formulas (10) and (11):

\[{\begin{vmatrix}1 & 1 & 1 & \ldots & 1\\1 & x & 1 & \ldots & 1\\\vdots & \vdots & \vdots & \ddots & \vdots\\1 & 1 & \cdots & \cdots & x \end{vmatrix}}_{n-1}= (x-1)^{n-1}={\begin{vmatrix}x & 1\\1 & 1\end{vmatrix}}^{n-1}\quad\quad(10)\]

and

\[{\begin{vmatrix}x & 1 & 1 & \ldots & 1\\1 & x & 1 & \ldots & 1\\\vdots & \vdots & \vdots & \ddots & \vdots\\1 & 1 & 1 & \ldots & x \end{vmatrix}}_{n}= + n(x-1)^{n-1} + (x-1)^{n}.\quad\quad(11)\]

Rubini then supplied the reader with three additional formulas, labeled (12), (13), and (14), which he never utilized further in his paper; he possibly wanted to show the reader the various results that the analytic theory of determinants permitted mathematicians to find.

Rubini's equation (15) is his formula (11) written completely in terms of determinants:

\[{\begin{vmatrix}x & 1 & 1 & \ldots & 1\\1 & x & 1 & \ldots & 1\\\vdots & \vdots & \vdots & \ddots & \vdots\\1 & 1 & 1 & \ldots & x \end{vmatrix}}_{n}={\begin{vmatrix}1 & 1 & 1 & \ldots & 1\\1 & x & 1 & \ldots & 1\\\vdots & \vdots & \vdots & \ddots & \vdots\\1 & 1 & 1 & \ldots & x \end{vmatrix}}_{n}+n\,{\begin{vmatrix}1 & 1 & 1 & \ldots & 1\\1 & x & 1 & \ldots & 1\\\vdots & \vdots & \vdots & \ddots & \vdots\\1 & 1 & 1 & \ldots & x \end{vmatrix}}_{n-1}.\quad(15)\]

The reader would have been able to follow this section more easily had Rubini presented equation (15) sequentially after formulas (10) and (11). Furthermore, it would have been of great assistance had Rubini gave the order of these matrices, because it is not apparent that these matrices are of different orders upon looking at them for the first time. For clarity in our explanation, equation (15) will be denoted as \(A_n = A_n' + n\,A_{n - 1}.\) The term \(A_n\) (which appears on the left-hand side of the equation) is an \(n\times n\) matrix with only \(x\)'s appearing along the main diagonal. The matrix \(A_n\) is a submatrix of the \((n+1) \times (n+1)\) matrix \(A_n'\) (which appears on the right-hand side of the equation); \(A_n'\) contains an additional row and column of ones. Rubini obtained the term \(n\,A_{n - 1}\) by substituting in formula (10); this is an \(n\times n\) matrix with \(n - 1\) \(x\)'s on the main diagonal. The reader would have also been better able to see the beauty of this equation through a numerical example, like the following one with \(x = 3\):

\[{\begin{vmatrix}3 & 1\\1 & 3\end{vmatrix}}_{2} = {\begin{vmatrix}1 & 1 & 1\\1 & 3 & 1\\1 & 1 & 3\end{vmatrix}}_{2} +2{\begin{vmatrix}1 & 1\\1 & 3\end{vmatrix}}_{1}.\]

The reader can check that each side of the equation equals \( 8.\)

This equation can be written as:

\[{\begin{vmatrix}3 & 1\\1 & 3\end{vmatrix}}_{2} - 2{\begin{vmatrix}1 & 1\\1 & 3\end{vmatrix}}_{1} = {\begin{vmatrix}1 & 1 & 1\\1 & 3 & 1\\1 & 1 & 3\end{vmatrix}}_{2}.\]

This version of the example illustrates that formula (15) provides the reader with a shortcut to the Laplace expansion for \(n\times n\) matrices of this form. It is unfortunate that part of this beauty is lost in Rubini's article due to the lack of structure and explanation of his formulas. Although several of the formulas he presented would have been clearer with some concrete examples, like the one supplied above, this lack may demonstrate Rubini's intention to impress rather than teach.

The article's lack of structure is also seen in Rubini's decision to merely present formulas (16) and (17), the only two formulas or equations of any kind provided in Section 4, and then to reproduce these formulas in slightly different form four sections later, at the beginning of Section 8. It makes one wonder why he decided to place these formulas in their own section and why he placed them so far in advance from the section in which he utilized them. He may have done this to emphasize the equations' importance, but it would have been easier for the reader to follow if he had presented these formulas for the first time in Section 8. The disorganized structure makes Rubini's work a bit difficult to follow, but also supports the conjecture that Rubini's work is a compilation of various mathematicians' previous works on determinants.