The purpose of the present article is to investigate the definition of matrix multiplication as a central issue in linear algebra courses. Applying both historical and pedagogical approaches, it focuses on the philosophy of generating the usual matrix multiplication, as a special binary operation, with its partly unexpected form compared with the…

In this paper, we reconstruct matrices from their minors, and give explicit formulas for the reconstruction of matrices of orders 2 × 3, 2 × 4, 2 × n, 3 × 6 and m × n. We also formulate the Plücker relations, which are the conditions of the existence of a matrix related to its given minors.

Given three planes in space, a complete characterization of their intersection is provided. Special attention is paid to the case when the intersection set does not exist of one point only. Besides the vector cross product, the tool of generalized inverse of a matrix is used extensively.

In this paper, we present a hypothetical learning trajectory (HLT) aimed at supporting students in developing flexible ways of reasoning about matrices as linear transformations in the context of introductory linear algebra. In our HLT, we highlight the integral role of the instructor in this development. Our HLT is based on the "Italicizing…

There are usually limited user evaluation of resources on a recommender system, which caused an extremely sparse user rating matrix, and this greatly reduce the accuracy of personalized recommendation, especially for new users or new items. This paper presents a recommendation method based on rating prediction using causal association rules.…

The singularity of a projective conic can be determined via the associated matrix to the implicit equation of the projective conic. In this expository article, we will first derive a known result for determining the singularity of a projective conic via the associated matrix. Then we will introduce the concepts of [mu]-basis of the parametric…

In this article, we derive one-parameter family of Schroder's method based on Gupta et al.'s (K.C. Gupta, V. Kanwar, and S. Kumar, "A family of ellipse methods for solving non-linear equations", Int. J. Math. Educ. Sci. Technol. 40 (2009), pp. 571-575) family of ellipse methods for the solution of nonlinear equations. Further, we introduce new…

Little is known about the variety of ways students conceptualize matrix multiplication, yet this is a fundamental part of most introductory linear algebra courses. My dissertation follows a three-paper format, with the three papers exploring conceptualizations of matrix multiplication from a variety of viewpoints. In these papers, I explore (1)…

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…

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.

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…

Hill ciphers are linear codes that use as input a "plaintext" vector [p-right arrow above] of size n, which is encrypted with an invertible n x n matrix E to produce a "ciphertext" vector [c-right arrow above] = E [middle dot] [p-right arrow above]. Informally, a near-field is a triple [left angle bracket]N; +, *[right angle bracket] that…

A lively example to use in a first course in linear algebra to clarify vector space notions is the space of square matrices of fixed order with its subspaces of affine, coaffine, doubly affine, and magic squares. In this note, the projection theorem is illustrated by explicitly constructing the orthogonal projections (in closed forms) of any…

In this paper, the properties of tangential and cyclic polygons proposed by Lopez-Real are proved rigorously using the theory of circulant matrices. In particular, the concepts of slippable tangential polygons and conformable cyclic polygons are defined. It is shown that an n-sided tangential (or cyclic) polygon P[subscript n] with n even is…