{"id":394,"date":"2019-03-12T08:51:19","date_gmt":"2019-03-12T08:51:19","guid":{"rendered":"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=394"},"modified":"2019-03-12T09:46:55","modified_gmt":"2019-03-12T09:46:55","slug":"introduction-to-matrices-and-matrix-algebra","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/chapter\/introduction-to-matrices-and-matrix-algebra\/","title":{"rendered":"Introduction to Matrices and Matrix Algebra"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n \r\n\r\n \r\n\r\nObjectives:<\/strong>\r\n\r\n \r\n\r\n1.\u00a0<\/strong>To familiarise the students with matrices and type of matrices.<\/strong>\r\n\r\n2.\u00a0<\/strong>To let the students carry out the basic operations of matrix algebra.<\/strong>\r\n\r\n \r\n\r\nIntroduction:<\/strong>\r\n\r\n \r\n

Matrix or Matrices (in plural) are used in many fields like graph theory, computer science, physics <\/span>and many more fields. For <\/span>example in graph <\/span>theory we use adjacency matrix for <\/span>finite graph. Each integer value in adjacency matrix represents the number of connection of <\/span>particular node. Rotation transformation can be represented in matrix form. Matrices are used to solve <\/span>linear system of equations. Matrices are used in cryptography. In cryptography large matrix is used to encode a message which code is difficult to break. <\/span>Receiver of <\/span>encoded message used <\/span>inverse of <\/span>matrix to decode the message. Earlier, in optics matrix algebra was used to record reflection and for refraction. Matrices are also used in <\/span>area of probability and statistics. For example, <\/span>probability vector which is also a matrix shows the probabilities of diverse outcomes of one experiment. So matrices are important in many fields.<\/span><\/p>\r\n \r\n

Matrices are also used in environmental sciences. For example, in <\/span>environmental impact assessment,<\/a> method, the <\/span>Leopold matrix<\/strong> is used as <\/span>tool. It measures the possible impact of a scheme on the <\/span>environment. <\/a>The component interaction matrix is used to explore <\/span>a estuarine system. <\/span>Similarly there are <\/span>many other <\/span>use of matrices in the field of environmental sciences.<\/span><\/p>\r\n \r\n

1.1<\/strong>\u00a0<\/span>Matrices<\/strong><\/p>\r\n \r\n

Definition: A matrix is a rectangular arrangement\/array of objects or elements. The horizontal arrays of a matrix are called its rows and the vertical arrays are called its columns.<\/span><\/p>\r\n\"\"\r\n

<\/div>\r\n
\r\n\r\n \r\n

is ?\u00d7? (to be read as \u2018m by n\u2019) matrix with ? rows and ? columns. In compact form the above matrix is represented by ?=[???<\/sub>]?\u00d7?<\/sub> or (???)?\u00d7?.<\/sub><\/p>\r\n \r\n

We will consider the elements as being real numbers and indicate an element by its row and column position.<\/span><\/p>\r\n \r\n

The???<\/sub>\u2032? are elements of the matrix ?, where ? indicates the row and ? indicates the column of the element. ?\u00d7? is its dimension, where ? is number of rows and ? is number of columns of the matrix.<\/p>\r\n \r\n\r\nExample 1.\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n<\/div>\r\n \r\n
\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n1.2<\/strong>\u00a0Special types of matrices:<\/strong>\r\n\r\n \r\n\r\n1.2.1<\/strong>\u00a0Null or zero matrix<\/strong>\r\n\r\n \r\n\r\nIf all elements of a matrix are zero then matrix is called null or zero matrix. It is denoted by ??\u00d7?<\/sub>.\r\nFor example:\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n1.2.2<\/strong>\u00a0Square matrix<\/strong>\r\n\r\n \r\n

A matrix is called square matrix if the number of rows of matrix are equal to the number of columns. A square matrix is said to have order if it has \u00d7 dimensions.<\/p>\r\n\r\n