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Journal articles on the topic 'Teaching of matrices and determinants'

Author: Grafiati

Published: 4 June 2021

Last updated: 4 February 2022

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1

Grattan-Guinness, I. "On proving certain optimisation theorems in plane geometry." Mathematical Gazette 97, no. 538 (March 2013): 75–80. http://dx.doi.org/10.1017/s0025557200005441.

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A pleasurable aspect of mathematics and its teaching is to review the diversity of ways in which theorems are proved. Especially in elementary branches, there are various kinds of proof: using (or avoiding) spatial geometry, analytic or coordinate geometry, common algebra, vectors, abstract algebras, matrices, determinants, the differential and integral calculus, and maybe mixtures thereof. Further, sometimes a proof of one kind is elegant while another is clumsy, or one proof of a theorem suggests why it follows while another proof is not perspicuous. There is also the question of whether a proof is direct or indirect (for example, proofby contradiction).

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2

FITRIAWAN, DONA. "PENGEMBANGAN BAHAN AJAR ALJABAR LINEAR ELEMENTER BERDASARKAN KEMAMPUAN KONEKSI MATEMATIS." Jurnal Pendidikan Matematika dan IPA 11, no. 2 (July 30, 2020): 217. http://dx.doi.org/10.26418/jpmipa.v11i2.37476.

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The purpose of this study is to develop: 1. elementary linear algebra teaching materials based on mathematical connection skills; 2. syllabus and lecture plan; 3. test mathematical connection skills. This type of research is a research and development approach whose research design consists of four stages, namely defining, planning, developing, and dissiminating. Data analysis techniques in this study describe narratively the steps in developing teaching materials. Based on the results of the analysis of the data obtained that: 1) the stages of developing teaching materials starting from the stages of defining, designing, until the first stage of development, namely expert validation. From this stage of development a revised elementary linear algebra teaching material has been produced based on input from three validators. Teaching materials compiled consist of four materials, namely systems of linear equations, matrices, inverses, and matrix determinants; 2) based on the opinions of three experts, elementary linear algebra teaching materials that have been compiled are classified as valid and good in terms of accuracy of contents, digestibility, use of language, so that they can be used to develop mathematical connection skills.AbstrakTujuan dari penelitian ini adalah untuk mengembangkan: 1. bahan ajar aljabar linier dasar berdasarkan keterampilan koneksi matematika; 2. silabus dan rencana kuliah; 3. menguji keterampilan koneksi matematika. Jenis penelitian ini adalah pendekatan penelitian dan pengembangan yang desain penelitiannya terdiri dari empat tahap, yaitu mendefinisikan, merencanakan, mengembangkan, dan menyebarluaskan. Teknik analisis data dalam penelitian ini menggambarkan secara naratif langkah-langkah dalam mengembangkan bahan ajar. Berdasarkan hasil analisis data diperoleh bahwa: 1) tahap pengembangan bahan ajar mulai dari tahap pendefinisian, perancangan, hingga tahap pertama pengembangan, yaitu validasi ahli. Dari tahap pengembangan ini bahan ajar aljabar linier revisi telah dihasilkan berdasarkan masukan dari tiga validator. Bahan ajar yang disusun terdiri dari empat bahan, yaitu sistem persamaan linear, matriks, invers, dan determinan matriks; 2) berdasarkan pendapat tiga ahli, bahan ajar aljabar linier dasar yang telah disusun diklasifikasikan sebagai valid dan baik dalam hal keakuratan isi, kecernaan, penggunaan bahasa, sehingga dapat digunakan untuk mengembangkan keterampilan koneksi matematis.Kata Kunci: bahan ajar, kemampuan koneksi matematis

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3

Silvester, John R. "Determinants of Block Matrices." Mathematical Gazette 84, no. 501 (November 2000): 460. http://dx.doi.org/10.2307/3620776.

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4

Maybee, John S., D. D. Olesky, Driessche P. van den, and G. Wiener. "Matrices, Digraphs, and Determinants." SIAM Journal on Matrix Analysis and Applications 10, no. 4 (October 1989): 500–519. http://dx.doi.org/10.1137/0610036.

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5

Reinhart, Georg Martin. "Determinants of Partition Matrices." Journal of Number Theory 56, no. 2 (February 1996): 283–97. http://dx.doi.org/10.1006/jnth.1996.0018.

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6

Basor, Estelle L., Yang Chen, and Harold Widom. "Determinants of Hankel Matrices." Journal of Functional Analysis 179, no. 1 (January 2001): 214–34. http://dx.doi.org/10.1006/jfan.2000.3672.

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7

Horáček, Jaroslav, Milan Hladík, and Josef Matějka. "Determinants of Interval Matrices." Electronic Journal of Linear Algebra 33 (May 16, 2018): 99–112. http://dx.doi.org/10.13001/1081-3810.3719.

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In this paper we shed more light on determinants of real interval matrices. Computing the exact bounds on a determinant of an interval matrix is an NP-hard problem. Therefore, attention is first paid to approximations. NP-hardness of both relative and absolute approximation is proved. Next, methods computing verified enclosures of interval determinants and their possible combination with preconditioning are discussed. A new method based on Cramer's rule was designed. It returns similar results to the state-of-the-art method, however, it is less consuming regarding computational time. Other methods transferable from real matrices (e.g., the Gerschgorin circles, Hadamard's inequality) are discussed. New results about classes of interval matrices with polynomially computable tasks related to determinant are proved (symmetric positive definite matrices, class of matrices with identity midpoint matrix, tridiagonal H-matrices). The mentioned methods were compared for random general and symmetric matrices.

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8

Chapman, Robin. "Determinants of Legendre symbol matrices." Acta Arithmetica 115, no. 3 (2004): 231–44. http://dx.doi.org/10.4064/aa115-3-4.

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9

Hilberdink, Titus. "Determinants of multiplicative Toeplitz matrices." Acta Arithmetica 125, no. 3 (2006): 265–84. http://dx.doi.org/10.4064/aa125-3-4.

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10

Kovacs, Istvan, Daniel S. Silver, and Susan G. Williams. "Determinants of Commuting-Block Matrices." American Mathematical Monthly 106, no. 10 (December 1999): 950. http://dx.doi.org/10.2307/2589750.

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11

Bruhn, Henning, and Dieter Rautenbach. "Maximal determinants of combinatorial matrices." Linear Algebra and its Applications 553 (September 2018): 37–57. http://dx.doi.org/10.1016/j.laa.2018.04.030.

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12

Kovacs, Istvan, Daniel S. Silver, and Susan G. Williams. "Determinants of Commuting-Block Matrices." American Mathematical Monthly 106, no. 10 (December 1999): 950–52. http://dx.doi.org/10.1080/00029890.1999.12005145.

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13

Molinari, Luca Guido. "Determinants of block tridiagonal matrices." Linear Algebra and its Applications 429, no. 8-9 (October 2008): 2221–26. http://dx.doi.org/10.1016/j.laa.2008.06.015.

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14

Stuart, Jeffrey L. "Determinants of Hessenberg L-Matrices." SIAM Journal on Matrix Analysis and Applications 12, no. 1 (January 1991): 7–15. http://dx.doi.org/10.1137/0612002.

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15

Tan, Yi-Jia. "Determinants of matrices over semirings." Linear and Multilinear Algebra 62, no. 4 (April 15, 2013): 498–517. http://dx.doi.org/10.1080/03081087.2013.784285.

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16

Gil', Michael. "Perturbations of determinants of matrices." Linear Algebra and its Applications 590 (April 2020): 235–42. http://dx.doi.org/10.1016/j.laa.2019.12.044.

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17

Robbins, David P., and Howard Rumsey. "Determinants and alternating sign matrices." Advances in Mathematics 62, no. 2 (November 1986): 169–84. http://dx.doi.org/10.1016/0001-8708(86)90099-x.

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18

Li, Zhongshan. "The determinants of GCD matrices." Linear Algebra and its Applications 134 (June 1990): 137–43. http://dx.doi.org/10.1016/0024-3795(90)90012-2.

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19

Krylov, P. A., and A. A. Tuganbaev. "Formal Matrices and Their Determinants." Journal of Mathematical Sciences 211, no. 3 (October 19, 2015): 341–80. http://dx.doi.org/10.1007/s10958-015-2610-3.

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20

Wang, Xiaoyuan. "Some determinants of Pascal-like matrices." Quaestiones Mathematicae 35, no. 2 (June 2012): 171–80. http://dx.doi.org/10.2989/16073606.2012.696839.

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21

Jeurissen, R. H. "On the determinants of divisor matrices." Discrete Mathematics 265, no. 1-3 (April 2003): 375–83. http://dx.doi.org/10.1016/s0012-365x(02)00880-4.

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22

Gander, Walter. "Zeros of Determinants of λ-Matrices." PAMM 8, no. 1 (December 2008): 10811–14. http://dx.doi.org/10.1002/pamm.200810811.

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23

Lindsay, Bruce G. "On the Determinants of Moment Matrices." Annals of Statistics 17, no. 2 (June 1989): 711–21. http://dx.doi.org/10.1214/aos/1176347137.

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24

Jantaramas, Trairat, Somphong Jitman, and Pornpan Kaewsaard. "Determinants of binomial-related circulant matrices." Special Matrices 6, no. 1 (July 1, 2018): 262–72. http://dx.doi.org/10.1515/spma-2018-0021.

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Abstract Due to their rich algebraic structures and wide applications, circulant matrices have been of interest and continuously studied. In this paper, n×n complex left and right circulant matrices whose first row consists of the coefficients in the expansion of (x + zy)n−1 are focused on, where z is a nonzero complex number and n is a positive integer. In the case where z ∈ {1, −1, i, −i}, explicit formulas for the determinants of such matrices are completely determined. Known results on the determinants of binomial circulant matrices can be viewed as the special case where z = 1. Finally, some remarks and open problems are discussed.

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25

Haake, Fritz, Marek Kus, Hans-Jürgen Sommers, Henning Schomerus, and Karol Zyczkowski. "Secular determinants of random unitary matrices." Journal of Physics A: Mathematical and General 29, no. 13 (July 7, 1996): 3641–58. http://dx.doi.org/10.1088/0305-4470/29/13/029.

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26

Gel'fand, I. M., and V. S. Retakh. "Determinants of matrices over noncommutative rings." Functional Analysis and Its Applications 25, no. 2 (1991): 91–102. http://dx.doi.org/10.1007/bf01079588.

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27

Bo, Zhou. "Determinants of Matrices on Partially Ordered Sets." Publikacije Elektrotehnickog fakulteta - serija: matematika, no. 13 (2002): 1–6. http://dx.doi.org/10.2298/petf0213001b.

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28

Matic, Ivan. "Inequalities with determinants of perturbed positive matrices." Linear Algebra and its Applications 449 (May 2014): 166–74. http://dx.doi.org/10.1016/j.laa.2014.02.026.

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29

Sothanaphan, Nat. "Determinants of block matrices with noncommuting blocks." Linear Algebra and its Applications 512 (January 2017): 202–18. http://dx.doi.org/10.1016/j.laa.2016.10.004.

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30

Basor, Estelle L., and Torsten Ehrhardt. "Some identities for determinants of structured matrices." Linear Algebra and its Applications 343-344 (March 2002): 5–19. http://dx.doi.org/10.1016/s0024-3795(01)00400-1.

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31

Fan, Jiangnan. "Determinants and multiplicative functionals on quaternion matrices." Linear Algebra and its Applications 369 (August 2003): 193–201. http://dx.doi.org/10.1016/s0024-3795(02)00722-x.

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32

Ballantine, Cristina M., Sharon M. Frechette, and John B. Little. "Determinants associated to zeta matrices of posets." Linear Algebra and its Applications 411 (December 2005): 364–70. http://dx.doi.org/10.1016/j.laa.2005.04.013.

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33

Moghaddamfar, A. R., S. Navid Salehy, and S. Nima Salehy. "The determinants of matrices with recursive entries." Linear Algebra and its Applications 428, no. 11-12 (June 2008): 2468–81. http://dx.doi.org/10.1016/j.laa.2007.11.021.

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34

Elsner, L. "Bounds for determinants of perturbed M-matrices." Linear Algebra and its Applications 257 (May 1997): 283–88. http://dx.doi.org/10.1016/s0024-3795(96)00174-7.

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35

Bussemaker, Frans, Irving Kaplansky, Brendan McKay, and Jacob Seidel. "Determinants of matrices of the conference type." Linear Algebra and its Applications 261, no. 1-3 (August 1997): 275–92. http://dx.doi.org/10.1016/s0024-3795(96)00412-0.

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36

Słowik, Roksana. "Inverses and Determinants of Toeplitz-Hessenberg Matrices." Taiwanese Journal of Mathematics 22, no. 4 (June 2018): 901–8. http://dx.doi.org/10.11650/tjm/180103.

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37

Basor, Estelle. "Asymptotics of determinants of block Toeplitz matrices." Random Matrices: Theory and Applications 06, no. 04 (October 2017): 1740003. http://dx.doi.org/10.1142/s2010326317400032.

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This paper is a survey of results that show how, in some cases, to compute the constant term which occurs in the Strong Szegö–Widom Limit Theorem for block Toeplitz matrices. While this constant in the scalar case has a long history, it is really only in a few instances that it can be calculated explicitly in the case of matrix-valued symbols.

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38

Cicuta, Giovanni M., and Madan Lal Mehta. "Probability density of determinants of random matrices." Journal of Physics A: Mathematical and General 33, no. 45 (November 3, 2000): 8029–35. http://dx.doi.org/10.1088/0305-4470/33/45/302.

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39

Mirashe, N., A. R. Moghaddamfar, S. H. Mozafari, S. M. H. Pooya, S. Navid Salehy, and S. Nima Salehy. "CONSTRUCTING NEW MATRICES AND INVESTIGATING THEIR DETERMINANTS." Asian-European Journal of Mathematics 01, no. 04 (December 2008): 575–88. http://dx.doi.org/10.1142/s1793557108000461.

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Let λ = (λi)i ≥ 1, μ = (μi)i ≥ 1, ν = (νi)i ≥ 1, ω = (ωi)i ≥ 1 and ψ = (ψi)i ≥ 1 be given sequences, and let (ai,j)i,j ≥ 1 be the doubly indexed sequence given by the recurrence [Formula: see text](i ≥ 3, j ≥ 2), with various choices for the two first rows a1,j, a2,j and first column ai,1. Note that the coefficients depend on the row index only. In this article we study the principal minors of doubly indexed sequences (ai,j)i,j ≥ 1 for certain sequences and certain initial conditions. Moreover, let (bi,j)i,j ≥ 1 be the doubly indexed sequence given by the recurrence [Formula: see text] with various choices for the first row b1,j and first column bi,1. We also study the principal minors of doubly indexed sequence (bi,j)i,j ≥ 1.

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40

Fasi, Massimiliano, and Gian Maria Negri Porzio. "Determinants of Normalized Bohemian Upper Hessenberg Matrices." Electronic Journal of Linear Algebra 36, no. 36 (June 13, 2020): 352–66. http://dx.doi.org/10.13001/ela.2020.5053.

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A matrix is Bohemian if its elements are taken from a finite set of integers. An upper Hessenberg matrix is normalized if all its subdiagonal elements are ones, and hollow if it has only zeros along the main diagonal. All possible determinants of families of normalized and hollow normalized Bohemian upper Hessenberg matrices are enumerated. It is shown that in the case of hollow matrices the maximal determinants are related to a generalization of Fibonacci numbers. Several conjectures recently stated by Corless and Thornton follow from these results.

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41

丁, 瀛帆. "Computation of the Determinants of Special Matrices." Advances in Applied Mathematics 08, no. 04 (2019): 716–30. http://dx.doi.org/10.12677/aam.2019.84082.

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42

Fiedler, Miroslav, Frank J. Hall, and Mikhail Stroev. "Permanents, determinants, and generalized complementary basic matrices." Operators and Matrices, no. 4 (2014): 1041–51. http://dx.doi.org/10.7153/oam-08-57.

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43

Krylov, P. A. "Determinants of Generalized Matrices of Order 2." Journal of Mathematical Sciences 230, no. 3 (March 13, 2018): 414–27. http://dx.doi.org/10.1007/s10958-018-3748-6.

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44

Shitov, Yaroslav. "The determinants of certain (0,1) Toeplitz matrices." Linear Algebra and its Applications 618 (June 2021): 150–57. http://dx.doi.org/10.1016/j.laa.2021.02.002.

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45

Orera, H., and J. M. Peña. "Accurate determinants of some classes of matrices." Linear Algebra and its Applications 630 (December 2021): 1–14. http://dx.doi.org/10.1016/j.laa.2021.07.020.

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46

Liu, Li, and Zhaolin Jiang. "Explicit Form of the Inverse Matrices of Tribonacci Circulant Type Matrices." Abstract and Applied Analysis 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/169726.

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It is a hot topic that circulant type matrices are applied to networks engineering. The determinants and inverses of Tribonacci circulant type matrices are discussed in the paper. Firstly, Tribonacci circulant type matrices are defined. In addition, we show the invertibility of Tribonacci circulant matrix and present the determinant and the inverse matrix based on constructing the transformation matrices. By utilizing the relation between left circulant,g-circulant matrices and circulant matrix, the invertibility of Tribonacci left circulant and Tribonaccig-circulant matrices is also discussed. Finally, the determinants and inverse matrices of these matrices are given, respectively.

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47

Moreno, I., M. M. Sánchez-López, C. Ferreira, J. A. Davis, and F. Mateos. "Teaching Fourier optics through ray matrices." European Journal of Physics 26, no. 2 (February 8, 2005): 261–71. http://dx.doi.org/10.1088/0143-0807/26/2/005.

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48

Moghaddamfar, A. R. "DETERMINANTS OF SEVERAL MATRICES ASSOCIATED WITH PASCAL'S TRIANGLE." Asian-European Journal of Mathematics 03, no. 01 (March 2010): 119–31. http://dx.doi.org/10.1142/s1793557110000088.

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Pascal's triangle is one of the most well-known arithmetical triangles and has many wonderful properties. This triangle may be rearranged so that one can consider various matrices. When these matrices are squares, we can discuss their determinants. Our purpose of this article is to study the determinants of square matrices related to Pascal's triangle where the determinants are equal to an entry in a particular place. We also consider the square matrices whose determinants are related to their dimensions only.

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49

Barwise, Amelia, and Mark Liebow. "Teaching Social Determinants of Health." Academic Medicine 95, no. 3 (March 2020): 329. http://dx.doi.org/10.1097/acm.0000000000003105.

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50

Hendrickson, Anders O. F. "Teaching Determinants Using Rook Arrangements." PRIMUS 28, no. 2 (January 9, 2018): 153–65. http://dx.doi.org/10.1080/10511970.2017.1350227.

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