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Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond revised 2018 September 16th University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 87 Lecture Outline Special Matrices Square, Symmetric, and Diagonal Matrices The Identity Matrix The Inverse Matrix Partitioned Matrices Permutations and Their Signs Permutations Transpositions Signs of Permutations Basic Lemma on Signs of Permutations Some Implications Determinants: Introduction Determinants of Order 2 Determinants of Order 3 The Determinant Function Permutation and Transposition Matrices Triangular Matrices University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 2 of 87 Outline Special Matrices Square, Symmetric, and Diagonal Matrices The Identity Matrix The Inverse Matrix Partitioned Matrices Permutations and Their Signs Permutations Transpositions Signs of Permutations Basic Lemma on Signs of Permutations Some Implications Determinants: Introduction Determinants of Order 2 Determinants of Order 3 The Determinant Function Permutation and Transposition Matrices Triangular Matrices University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 3 of 87 Square Matrices Asquare matrix has an equal number of rows and columns, this number being called its dimension. The (principal, or main) diagonal of a square matrix A = (a ) of dimension n n ij n×n is the list (a ) =(a ,a ,...,a ) of its n diagonal elements. ii i=1 11 22 nn The other elements a with i 6= j are the off-diagonal elements. ij Asquare matrix is often expressed in the form a a . . . a 11 12 1n a a . . . a A= 21 22 2n . . . . . . .. . . . . a a . . . a n1 n2 nn with some extra dots along the diagonal. University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 4 of 87
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