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The multiplication of a vector \(\vec{v_1}\) by a square matrix produces another vector of the same dimensions of \(\vec{v_1}\). For example, we can multiply a \(2\times 2\) matrix and a 2-dimensional vector: \[\begin{pmatrix} a&b \\ c&d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}=\begin{pmatrix} ax+by \\ cx+dy \end{pmatrix} \nonumber \]
- Using Matrices to Balance Chemical Reactions and Modeling the Implications of a Balanced Reaction
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- MATHEMATICAL DESCRIPTION AND SOLUTION
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Emilee Barrett University of South Florida Advisors: Arcadii Grinshpan, Mathematics and Statistics Frankie Costanza, Chemistry Problem Suggested By: Frankie Costanza Follow this and additional works at: https://digitalcommons.usf.edu/ujmm Part of the Mathematics Commons UJMM is an open access journal, free to authors and readers, and relies on your...
This paper explores an alternative way to balancing equations of chemical reactions and understanding why it is necessary to use balanced equations in science. To balance a chemical equation chemists often guess the coeficients that would balance the equation and check if it works. An alternative method is to use matrices to determine the coeficien...
balanced chemical equations, matrix operations, systems of linear Diophantine equations, atmospheric pollution
Investigating how to use matrices to determine the stoichiometric values of chemical reactions and how to use a balanced chemical reaction and predetermined reactant or product quantities that can produce a model.
Matrices are used to find the coefficients (integer numbers) of multiple variables in a system of Diophantine linear equations (Lazebnik) provided that the variables and system create the square matrices. For simplicity, we consider the invertible matrices. Creating the suitable invertible matrix and vector, finding the inverse and determinant of t...
To appropriately use a chemical equation, it is imperative for it to be balanced correctly. If the ratio between the species is inaccurate then what is produced may be underestimated or overestimated which is not viable in scientific investigations. In the case of environmental pollution, if scientists only write chemical equations as a 1:1 ratio t...
Chemical reactants help people understand how certain molecules are produced and in what quantities. It also identifies how much of the reactants are needed to carry out the reaction. Balancing the chemical reaction specifies these two identities of a chemical reaction and aids in understanding how to prevent or increase the process. The chemical e...
- Emilee Barrett
- 2019
Nov 6, 2013 · A matrix (transformation matricies) may be used to change the scale of an object, to model reflections across lines and points, find new or old locations after rotation of coordinate axes, and even do some simple derivatives in calculus.
The use of matrix equations to solve arbitrarily large problems leads to a field of mathematics called linear algebra.
Mar 17, 2024 · Throughout this post, we’ve explored what stoichiometry is, provided clear examples, and offered practice problems to help you build your skills. Remember, the key to mastering stoichiometry lies in practice. The more you work through problems, the more intuitive these concepts will become.
A square matrix \(\arr{A}\) is invertible or non-singular if there is a matrix \(\arr{B}\) such that \[\begin{align*} \arr{A}\arr{B}=\arr{B}\arr{A}=\arr{I} \end{align*}\] Ex: \(\arr{A}=\begin{bmatrix}2&4\\3&1\end{bmatrix}\) and \(\arr{B}=\begin{bmatrix}-1/10&2/5\\3/10&-1/5\end{bmatrix}\) are inverses of each other because \(\arr{A}\arr{B}=\arr ...
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Chapter Objectives. Learn the nomenclature used in linear algebra to describe matrices (rows, columns, triangular matrices, diagonal matrices, trace, transpose, singularity, etc). Learn how to add, subtract and multiply matrices. Learn the concept of inverse. Understand the use of matrices as symmetry operators.
