$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$, 2.5: Application of Matrices in Cryptography, [ "article:topic", "license:ccby", "showtoc:no", "authorname:rsekhon" ], $$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$, 2.5.1: Application of Matrices in Cryptography (Exercises). 2. a square matrix of order 3, the message is cut into pieces (WEL), (COM), (E), each of length 3, and \end{array}\right] \quad \bf{(I)} \nonumber\], Now to encode, we multiply, on the left, each matrix of our message by the matrix $$A$$. By 1900, they were just a finite dimensional sub case of the emerging theory of linear transformations. 11 \\ square matrix is in cryptography. 14 1 \end{array}\right]\left[\begin{array}{c} 20 \\ 2. can also be used to represent real world data like the population of people, infant mortality rate, etc. Leibnitz, one of the founder of calculus, used determinant in 1963 and Cramer presented his determinant based formula for solving systems of linear equation (today known as Cramer’s rule) in 1750. 15 \\ 43 For example ,the matrix A mentioned above  is a 3*4 matrix ,where 1,5,9,2,6 etc are its elements. Matrices have many applications in diverse fields of science, commerce and social science. 3. (decrypts) the message to retrieve the original message by using the inverse of Lagrange desired to characterize the maxima and minima of multi-variant functions. Later the physicist, P.A.M. Dirac introduced the term “bracket” for what we now call the “scalar product” of a “bar” (row) vector times a “ket” (column) vector and the term “ket-bra” for the product of a ket times a bra, resulting in what we now call a simple matrix, as above. In mathematics, one application of matrix notation supports graph theory. \mathrm{C} \\ In this section we will examine a method of encryption that uses matrix multiplication and matrix inverses. Alan Turing and Von Neumann, the 20th century giants in the development of stored – program computers. 1 \\ \[\mathrm{B}^{-1}=\left[\begin{array}{ccc} \end{array}\right]\left[\begin{array}{c} A matrix is defined as a rectangular array of numbers or symbols which are generally arranged in rows and columns. Cayley studied multiplication so that the matrix of coefficient for the composite transformation ST is the product of the matrix S times the matrix T. He went on to study the algebra of these composition including matrix inverses. :  Keep in mind that the order in which matrices are added is not important; thus, we can say that  A + B is equal to B + A.

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