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Unformatted text preview: linear equations: Ax = 0 (4) (a) Suppose A is nonsingular. Show that there is a unique solution to the system of equations (4). (b) Suppose A is singular. Show that there are an inﬁnite number of distinct solutions to the system of equations (4). 5. [Determinant of Upper Triangular Matrix] Let A be an n × n matrix, with a ij = 0 whenever i > j. (a) Show that: det A = Y n i =1 a ii (b) Use (a) to verify that A is nonsingular if and only if a ii 6 = 0 for each i ∈ { 1 ,...,n } . 6. [Test of Linear Dependence of Vectors] Let S = { x 1 ,x 2 ,...,x m } be a set of vectors in R n , and let G be the m × m matrix deﬁned by: G = x 1 x 1 ··· x 1 x m . . . ··· . . . x m x 1 ··· x m x m where x i x j is the inner product of x i and x j , for i = 1 ,...,m and j = 1 ,...,m. Show that S is linearly dependent if and only if: det G = 0 2...
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This note was uploaded on 10/03/2009 for the course ECON 6170 taught by Professor Mitra during the Fall '08 term at Cornell.
 Fall '08
 MITRA

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