Recorrências lineares, isometria, criptografia e outras aplicações envolvendo matrizes 2 por 2

Abstract

The present study has as its main theme to show the applications involving square matrices of order 2. To achieve the objective it is showed the definition of matrices, the operations and its properties as well as the study of transposed and invertible matrix and determinant calculation being restrict to matrices of order 2. After, we define isometrics in plain as a geometric transformation that preserves distance and angles. We introduce the rotation, translation and reflection matrix presentation and insert that all isometry is ƒ (u) = T(u)+w, where T is an orthogonal linear application. We define similar matrices and their properties finding enough and necessary conditions so that a square matrix of order 2 can be diagonalizable, as well as the corresponding diagonal matrix and the conjugate matrix. We’ve calculated the nth power of a square matrix of order 2 and then we’ve solved linear relations of recurrence expressed as xn+1 = axn+bxn-1, particularly Fibonacci sequence. We’ve studied the conics represented by the equation ax2+2bxy+cy2+dx+ey+ƒ=0, where through isometries we identified as being, ellipse, hyperbola, parabola, point, line, a pair of parallel lines or concurrent and even empty set. We’ve ended the study with a cryptography using matrices multiplication and the calculation of invertible matrices.