Page: 3 1.0 Aims and objectives In this lesson, we have discussed the representation of data using matrices and the representation of expressions using determinant. Matrix inversion, a convenient way to solve system of linear equations, is also discussed in this lesson. After reading this lesson, you should able to understand · Matrix and its usage in data representation. · Determinant and the various methods of computing it. · Cofactors and Adjoint of a matrix. · How to find matrix Inverse? And to solve linear equations using inverse. · Properties of determinant and invertible matrices. · Applications of Matrix inverses in real-time simulations 1.1 Introduction J. J. Sylvester coined the term “matrix” in 1848. A matrix is a rectangular table of elements. Matrices are used to describe linear equations, keep track of the coefficients of linear transformation and to record data that depend on multiple parameters. The determinant notation is now employed in almost every branch of applied science. Matrix inversion plays a significant role in computer graphics and to solve system of linear equations.
Page: 4 1.2 Matrices The study of matrices is quite old. A 3-by-3 magic square appears in Chinese literature dating from as early as 650 BC. Matrices have a long history of application in solving linear equations. After the development of the theory of determinants by Seki Kowa and Leibniz in the late 17th century, Cramer developed the theory further in the 18th century, presenting Cramer's rule in 1750. Carl Friedrich Gauss and Wilhelm Jordan developed Gauss-Jordan elimination in the 1800s. Cayley, Hamilton, Grassmann, Frobenius and von Neumann are among the famous mathematicians who have worked on matrix theory. Olga Taussky-Todd (1906-1995) used matrix theory to investigate an aerodynamic phenomenon called fluttering or aeroelasticity during WWII. In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. Matrices are used to describe linear equations, keep track of the coefficients of linear transformation and to record data that depend on multiple parameters. Matrices can be added, multiplied, and decomposed in various ways, making them a key concept in linear algebra and matrix theory. 1.2.1 Applications of matrices Ø Encryption Matrices can be used to encrypt numerical data. Multiplying the data matrix with a key matrix does encryption. Simply multiplying the encrypted matrix with the inverse of the key does decryption. Ø Computer graphics 4×4 transformation matrices are commonly used in computer graphics. The upper left 3×3 portion of a transformation matrix is composed of the new X, Y, and Z-axes of the post-transformation coordinate space. 1.2.2 Definition of a Matrix If m and n are positive integers, then an m ´ n matrix (read “m b y n”) is a rectangular array æ a11 a12 a13 . . a1n öü ç ÷ï ç a 21 a 22 a 23 . . a 2 n ÷ï ça a32 a33 . . a3n ÷ïï ç 31 ÷ý m rows . . . . . ÷ï ç . ç . . . . . . ÷÷ï ç ï ça a n 2 a n 3 . . a nn ÷øïþ n1 è14 4444244444 3 n columns in which each entry, ai,j, of the matrix is a real number. An m ´ n matrix has m rows (horizontal lines) and n columns (vertical lines).
Page: 5 A matrix having m rows and n columns is said to be of order m ´ n. If m = n, the matrix is square of order n. For a square matrix, the entries a11, a22, a33,…, ann are the main diagonals. A matrix that has only one row is a row matrix, and a matrix that has only one column is a column matrix. 1.2.3 Matrix types 1) Diagonal Matrix A square matrix A is said to be a diagonal matrix if aij= 0 when i ¹ j. In a diagonal matrix all the entries except the entries along the main diagonal is zero. 2) Triangular matrix A square matrix in which all the entries above the main diagonal are zero is called a lower triangular matrix. If all the entries below the main diagonal are zero, it is called an upper triangular matrix. 3) Scalar Matrix A scalar matrix is a diagonal matrix in which all the entries along the main diagonal are equal. 4) Identity Matrix or Unit Matrix An Identity Matrix or Unit Matrix is a scalar matrix in which entries along the main diagonal are equal to 1. 5) Zero or Null or Void Matrix In which all the entries are zero. 6) Equality of matrices The matrices A=[aij]mxn and B=[bij]pxq are equal if m = p, n = q and aij = bij for every i and j. 1.3 Determinant Many complicated expressions of electrical and mechanical systems can be conveniently handled by expressing them in “determinant form”. 1.3.1 Definition In algebra, a determinant is a function depending on n that associates a scalar, det(A) or D, to every n × n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra.
Page: 6 (or) The mathematical expression a1 b2 -a2 b1 can be expressed as a1 b1 a 2 b2 The above form is called a determinant of the second order. Each of the numbers a1, b1 , a2, b2 is called an element of the determinant. Thus 2 5 3 9 = (2)(9) - (3)(5) = 18 - 15 = 3. Similarly, the arrangement of nine elements as a 1 b 1 c1 a 2 b2 c2 a 3 b3 c3 is called a determinant of the third order. Thus 3 1 3 2 2 - 1 = 3(4 - 3) - 1(4 + 1) + 3(-6 - 2) = 3 - 5 - 24 = -26 1 -3 2 In general, a determinant of the nth order is denoted by a1 a2 b1 b2 . . l1 . . l2 - - - - - - - -(1) . . . . . . . . . . a n bn . . l n The determinant in (1) have n rows and n columns; thus having n2 elements. The diagonal through the left hand top corner which contains the elements a1 , b2 , c3 , ---, ln is called the leading or principal diagonal.