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Determinants and matrices Definition : A matrix is a rectangular array of numbers Hor functionsL arranged in rows and columns Example : A=J 2 1 3 N 1 0 2 is a 2 ´ 3 H2 by 3L matrix because it has 2 rows and 3 columns. A m ´ n matrix is said to be of order m ´ n. H1 2 3 L, order 1 ´ 3 jij 6 zyz, order 2 ´ 1 Column Matrix : k1{ Square Matrix : A matrix with same number of rows as columns * Row Matrix : Examples : J ex a b j N, i j 2x c d ke 3 xy z z x2 { are 2 ´ 2 square matrices OR square matrices of order 2. The numbers Hor functionsL are called "entries" or elements " of the matrix. Double suffix notation : Consider m ´ n matrix A a11 i j j j a21 j j A=j j j j j . k am1 a12 a22 . am2 . a1 n y z z . a2 n z z z z z . . z z z . amn { The element in ith row and j th column is denoted by aij where i = row number j = column number Determinants : Example : det J Eg : Ë a11 a21 With each square matrix we associate a number denoted by det HaL or È aij È or È A È called the determinant of A Determinant of order 2 a12 a N = Ë 11 a22 a21 a12 Ë = a11 a22 - a21 a12 a22 3 -2 Ë = 3 ´ 5 - 4 ´ H-2L = 15 + 8 = 23 4 5 H a number L Example : Construct a 2 nd order det in which aij = H1 + 3 iL - j2 and evaluate. Clearly i = 1, 2, a11 a12 a21 a22 j = 1, 2, thus = H1 + 3L - 12 = 4 - 1 = 3 = H1 + 3L - 22 = 4 - 4 = 0 = H1 + 3 ´ 2L - 12 = 7 - 1 = 6 = H1 + 3 ´ 2L - 22 = 7 - 4 = 3 det Haij L = Ë 3 0 Ë = 3´3 - 6´0 = 9 6 3 Minors and Cofactors : The minor Mij associated with element aij in an n th order det is defined to be a det of order n - 1 obtained by removing ith row and jth column of the original det. Example : a11 i j j j j a21 j j j k a31 The minor of a22 is Ë a12 a13 y z z z a22 a23 z z z z a32 a33 { a13 Ë a33 a11 a31 Laplace Exapansion of Dets : We know D = Ë a11 a21 a12 Ë = a11 a22 - a21 a12 a22 Here c11 = a22 , c12 = a12 , c21 = -a12 , c22 = a11 Clearly D = a11 c11 + a12 c12 HExpansion by first rowL Similarly D = a11 c11 + a21 c21 D = a21 c21 + a22 c22 D = a12 c12 + a22 c22 H Expansion by first column L H Expansion by 2 nd row L H Expansion by 2 nd columnL