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Unformatted text preview: LINEAR MODELS AND MATRIX ALGEBRA Continued Chapter 5 Alpha Chiang, Fundamental Methods of Mathematical Economics 3 rd edition Conditions for Nonsingularity of a Matrix When squareness condition is already met, a sufficient condition for the nonsingularity of a matrix is that its rows be linearly independent. Squareness and linear independence constitute the necessary and sufficient condition for nonsingularity. Nonsingularity squareness and linear independence Conditions for Nonsingularity of a Matrix An n x n coefficient matrix A can be considered as an ordered set of row vectors, ie., as a column vector whose elements are themselves row vectors: where For the rows to be linearly independent, none must be a linear combination of the rest. [ ] 11 12 1 1 21 22 2 2 1 1 1 1 2 ' ' ' where ' , 1,2, , n n n n n n i i i in a a a v a a a v A a a a v v a a a i n = = = = L L L L L L M L L L Conditions for Nonsingularity of a Matrix Example Since [6 8 10] = 2 [3 4 5], We have v 3 = 2 v 1 + 0v 2 . Thus the third row is expressible as a linear combination of the first two, the rows are not linearly independent. ' 1 ' 2 ' 3 3 4 5 1 2 6 8 10 v A v v = = Rank of a Matrix Rank r = the maximum number of linearly independent rows that can be found in a matrix. It also tells us the maximum number of linearly independent columns in the said matrix. By definition, an n x n nonsingular matrix has n linearly independent rows (or columns); consequently, it must be of rank n. Conversely, an n x n matrix have a rank n must be nonsingular. Determinants and Nonsingularity The determinant of a square matrix A, denoted by A, is a uniquely defined scalar (number) associated with that matrix. Determinants are uniquely defined only for square matrices. For a 2 x 2 matrix , its determinant is defined to be the sum of two terms as follows: [= a scalar] 11 12 21 22 a a A a a = 11 12 11 22 21 12 21 22 a a A a a a a a a = = Example: their determinants are 10 4 3 5 8 5 1 Given A and B = =  10 4 10(5) 8(4) 18 8 5 3 5 3( 1) 0(5) 3 1 A B = = = = = =  Relationship between linear dependence of rows in matrix A vs....
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 Spring '11
 richards
 Economics

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