application of matrices
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Application of Matrices
Presented By:Mohammedi Limdiwala
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Matrix
▪ A matrix is a rectangular arrangement of ‘mn’ elements.▪ It consists of rows and columns arrangements.▪ It is a systematic arrangement of elements and may
represent vectorial or scalar quantity.
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Some of the applications of Matrix
▪ Engineering forces on a bridge or truss▪ Electronics▪ Genetics (working out selection process)▪ Probability (finding quantities in a chemical reaction)▪ Chemistry▪ Economics (study of stock market, etc)▪ Encryption And decryption messages –
Cryptography(here we will discuss Cryptography)
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Encryption and Decryption (Cryptography)
▪ A cryptogram is a message written according to a secret code (the Greek word kryptos means “hidden”). The following describes a method of using matrix multiplication to encode and decode messages.
▪ Then convert the message to numbers and partition it intoencoded row matrices, each having entries In the following example we take an example to encode the Message “MEET ME MONDAY”
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Encoding Process
▪ Partitioning the message (including blank spaces, but ignoring punctuation) into groups of three produces the following uncoded row matrices.
▪ To encode a message, choose an invertible matrix and multiply the uncoded row matrices (on the right) by to obtain coded row matrices. This inverted matrix will act as a password.
▪ For example here.
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Encoding Process
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Encoding Process
▪ The sequence of coded row matrices is
▪ Finally, removing the matrix notation produces the following cryptogram
▪ For those who do not know the encoding matrix A, decoding the cryptogram is possible.
▪ But for an authorized receiver who knows the encoding matrix decoding is relatively simple. The receiver just needs to multiply the coded row matrices by A-1 to retrieve the uncoded row matrices. In other words, if
▪ is an uncoded 1×n matrix, then Y=XA is the corresponding encoded matrix. The receiver of the encoded matrix can decode by Y multiplying on the right by A-1 .
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Decoding Process
▪ Begin by using Gauss-Jordan elimination to find A-1
▪ Now, to decode the message, partition the message into groups of three to form the coded row matrices
▪ To obtain the decoded row matrices, multiply each coded row matrix by A-1
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Decoding Process
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Decoding Process
▪ The sequence of decoded row matrices is
▪ and the message is
▪ And the authorized person can easily decode the encoded message using the inverse of the matrix A, which was the password during encoding.
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