TIMBILA BOYS HIGH SCHOOL T 0 PIC MATRIX

TIMBILA BOYS HIGH SCHOOL. T 0 PIC : MATRIX PBL PRESENTATION. FORM THREE NORTH GROUP B.

GROUP MEMBERS. 1. MUSA DANIEL 2. JOSEPH DENA 3. DONALD MRABAI 4. RASHID ANTONY 5. MIKAL RAMA 6. PAUL MROSSO

1. What is matrix? Is a two-dimensional arrangement of numbers in rows and columns enclosed by a pair of square brackets ([]). . It is also an organization structure in which two or more lines of command, responsibility or Communication may run through the same individual. . The size, or dimension, of the matrix is n x m, where, . n is the number of rows of the matrix, . m is the number of column of the matrix.

. For example, the following matrices are of dimensions 1 x 4, 3 x 1, 2 x 3 and 3 x 2 respectively. (1 -1 0 4) 2 012 -4 9 -2 -3 0 2 27 0 3 0 A special kind of matrix is a square matrix, I. E , a matrix with the same number of rows and columns If a square matrix has n rows and n columns , we say that matrix has order n The matrix 1 2 0 is square matrix of order 3. 0 -1 3 2 4 5

TYPES OF MATRICES Row matrix Column matrix / vector matrix Zero matrix / null matrix Diagonal matrix Singular matrix Scalar matrix Unit matrix Upper triangular matrix Lower triangular matrix NB; in mathematics , a matrix is a rectangular array of numbers, symbols , or expression arranged in rows and columns

IMPORTANCE OF MATRIX It enables to identify the specific person with the idea of rows and columns eg; in a class or in a hospital. It helps in calculating population in a particular institution eg ; school It enables architectures to get /earn income from matrix by marix construction. It provides an easy of offloading and loading of containers in cargos. It helps in balancing in an aeroplane to ensure stability of the aeroplane.

APPLICATIONS OF MATRICES Graph theory The adjacency matrix of a finite graph is notion of graph theory. Linear combination of quantum state in physics The first model of quantum mechanics by Heisenberg in 1925 represented theory's operator by infinite – dimensional matrix acting on quantum states. This is also referred to as matrix mechanics Computer graphic 4 x 4 transformation rotation matrix are commonly used in computer graphics Solving linear equations using row reduction crammer's rule (determinants) using the inverse matrix

CRYPTOGRAPHY Is concerned with keeping in communications private Cryptography mainly consist encryption and decryption Encryption is the transformation of data into some unreadable form Its purpose its to ensure privacy by keeping the information hidden from anyone for whom it is not intended , even those who can see the encrypted data Decryption is the reverse of encryption it is the transformation of encrypted data back into some intelligible form Encryption and decryption require the use of some secret information usually referred to as a key Depending on the encryption mechanism used , the same key might be used for both encryption and decryption , while for other mechanisms , the keys used for encryption and decryption might be different

APPLICATION OF MATRIX TO CRYPTOGRAPHY One type of code , which extremely difficult to break , makes use of a large matrix to encode a message The receiver of the message decodes it is using the inverse of the matrix. The first matrix used by the sender is called the encoding matrix and its inverse is called the decoding matrix, which is used by the receiver. MESSAGE TO BE SENT: PREPARE TO NEGOTIATE The encoding matrix be -3 -3 -4 0 1 1 4 3 4

We assign a number for each letter of the alphabet. such that A is 1 , B is 2 and so on. Also , we assign the number 27 to space between two words. That’s the message becomes. P R E P A R E * T O * N E G O T I A T E 16 18 5 16 1 18 5 27 20 15 27 14 5 7 15 20 9 1 20 5

ENCODING Since we are using a 3 by 3 matrix , we break the enumerated message above into a sequence of 3 by 1 vectors 16 16 5 18 1 27 27 5 18 20 14 15 5 20 20 7 9 5 15 1 27 NB; it was necessary to add a space at the end of the message to complete the last vector. We encode the message by multiplying each of the above vectors by the encoding matrix. We represent above vector as columns of a matrix and perform its matrix multiplication with the encoding matrix.

-3 -3 -4 0 1 1 4 3 4 16 16 5 15 5 20 20 18 1 27 27 7 9 5 5 18 20 14 15 1 27 We get; -122 -123 23 19 138 139 -176 -182 -96 -91 -183 47 41 22 10 32 181 197 101 111 203 The columns of this matrix gives the encoded message. Encoding is complete

TRANSMISSION The message is transmitted In a linear form -122 23 138 -123 -182 41 197 -183 32 203 -96 19 139 -176 22 101 47 -91 10 181 101

DECODING TO DECODE THE MESSAGE The receiver writes this string as sequence of 3 by 1 column matrix and repeats the technique using the inverse of the encoding matrix. The inverse of this encoding is the decoding matrix 1 0 1 4 4 3 -4 -3 -3 To decode the message, to perform the matrix multiplication 1 0 1 -122 -123 4 4 3 23 19 -4 -3 -3 138 139 -176 -182 -96 -91 -183 47 41 22 10 32 181 197 101 111 203

Matrix obtained is; 16 16 5 15 5 20 20 18 1 27 27 7 9 5 5 18 20 14 15 1 27

DECODED MESSAGE The columns of this matrix, written in linear form, give the original message 16 18 5 16 1 18 5 27 20 15 27 14 5 7 15 20 9 1 20 5 P R E P A R E. T O. N E G O T I A T E Message received: PREPARE TO NEGOTIATE END OF OUR PRESENTATION. BYE.

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