James Sylvester (his original name was "James Joseph." He adopted the new family name "Sylvester" following his brother who lived in the US under the name "Sylvester") was born into a Jewish family in London, and was to become one of the supreme algebraists of the nineteen century. At the age of 14, Sylvester was a student of Augustus De Morgan at the University of London. His family withdrew him from the University after he was accused of stabbing a fellow student with a knife. Despite having studied for several years at St John's College, Cambridge, he was not permitted to take his degree there because he "professed the faith in which the founder of Christianity was educated." Therefore, he received his degrees from Trinity College, Dublin.

In 1841, James moved to the United States to become a professor of mathematics at the University of Virginia, but left after less than four months following a violent encounter with two students he had disciplined. He moved to New York City and began friendships with the Harvard mathematician Benjamin Peirce and the Princeton physicist Joseph Henry. However, he left in November 1843 after being denied appointment as Professor of Mathematics at Columbia College (now University), again for his Judaism, and returned to England. He was hired in 1844 by the Equity and Law Life Assurance Society for which he developed successful actuarial models and served as de facto CEO, a position that required a law degree. In 1872, he finally received his B.A. and M.A. from Cambridge, having been denied the degrees due to his being a Jew. In 1876 Sylvester again crossed the Atlantic Ocean to become the inaugural professor of mathematics at the new Johns Hopkins University in Baltimore, Maryland. Sylvester was an avid poet, prefacing many of his mathematical papers with examples of his work.

A matrix (plural matrices) is a rectangular array of numbers, functions, or any symbols. It can be written as

We denote this array by a single letter A (usually a capital boldfaced letter) or by \( \left( a_{i,j} \right) \) or \( \left[ a_{i,j} \right] , \) depending what notation (parenthesis or brackets) is in use. The symbol \( a_{i,j} ,\) or sometimes \( a_{ij} ,\) in the ith row and jth column is called the \( \left( i, \, j \right) \) entry. We say that A has m rows and n columns, and that it is an \( m \times n \) matrix or m-by-n matrix, while m and n are called its dimensions. The double subscript notation for the elements permits us to discuss the elements of entries of a general matrix in an efficient manner. Note that the number of rows is always listed first, while the second subscript on the entry indicates the column in which it occurs. The ordered pair (i, j) is called the address of the entry 𝑎_{i, j}, and the entry in the (i, j) position of A is 𝑎_{i, j}. We also refer to A as a matrix of size m × n. Matrices with a single row are called row vectors, and those with a single column are called column vectors.

Now we give a formal definition of a matrix, which is not general, but it is sufficient for this part of the tutorial.

Mathematica also uses the more general term "Array."

Here is an example of 4×6 matrix from ℤ^4×6:

Before we can discuss arithmetic operations for matrices, we have to define equality for matrices. Two matrices are equal if they have the same size and their corresponding elements are equal. A matrix with elements that are all 0's is called a zero or null matrix. A null matrix usually is indicated as 0.

As we will see, a matrix can represent many different things. However, in this part, we will focus on how matrices can be used to represent systems of linear equations. Any \( m \times n \) matrix can be considered as an array of \( n \) columns

Before we can discuss arithmetic operations for matrices, we have to define equality for matrices. Two matrices are equal if they have the same size and their corresponding elements are equal. A matrix with elements that are all 0's is called a zero or null matrix. A null matrix usually is indicated as 0.

Matrices are fundamental objects in linear algebra, so there are a variety of ways to construct a matrix in Mathematica. Generally, you need to specify what types of entries the matrix contains (more on that to come), the number of rows and columns, and the entries themselves. First, let's dissect an example:

Now we are going to introduce matrix multiplication that may at first seem rather strange. We don't know exactly by whom or when the multiplication of matrices was invented. At least we know that the work of 1812 by Jacques Philippe Marie Binet (1786--1856) contains the definition of the product of matrices. Jacques Binet was a French mathematician, physicist and astronomer who made significant contributions to number theory, and the mathematical foundations of matrix algebra which would later lead to important generalizations by Cayley, Sylvester, and others. In his memoir on the theory of the conjugate axis and of the moment of inertia of bodies he enumerated the principle now known as Binet's theorem.

When two linear maps are represented by matrices, then the matrix product represents the composition of the two maps. This concept is due to the German mathematician Gotthold Eisenstein (1823--1852), a student of Carl Gauss. Despite that Gotthold was born to the Jewish family, they converted from Judaism to become Protestants. His father served in the Prussian army for eight years. Gotthold mathematical talents were recognized early and he became a student of Carl Gauss. Gotthold also showed a considerable talent for music from a young age and he played the piano and composed music throughout his life. Eisenstein introduced the notation \( {\bf A} \times {\bf B} \) to denote the matrix multiplication around 1844. The idea was then expanded on and formalized by Carley in his Memoir on the Theory of Matrices that was published in 1858. However, Gotthold, who survived childhood meningitis, suffered bad health his entire life.