Problem_Set_4

# Problem_Set_4 - linear equations Ax = 0(4(a Suppose A is...

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T.Mitra, Fall, 2008 Economics 6170 Problem Set 4 [Due on Wednesday, October 8] 1. [System of Linear Equations: Existence and Uniqueness of Solutions] Consider the following system of linear equations: 2 x 1 + 4 x 2 = 8 3 x 1 + 3 x 2 = 9 2 x 1 + 3 x 2 = 7 (1) (a) Show, using the existence criterion discussed in class, that the system of equations (1) has a solution. (b) Does the system of equations (1) have a unique solution ? Explain. 2. [System of Linear Equations: Existence of Solutions] Consider the following system of linear equations: 3 x 1 + x 2 + x 3 = t x 1 - x 2 + 2 x 3 = 1 - t (2) x 1 + 3 x 2 - 3 x 3 = 1 + t where t is a real number. For what values of t will the system of equations (2) have a solution ? Explain. 3. [System of Linear Equations: Uniqueness of Solution] Let A be an m × n matrix and let b be a vector in R m . Consider the following system of linear equations: Ax = b (3) Suppose (3) has a unique solution for every b R m . Can m be diﬀerent from n ? Explain. 1

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4. [System of Homogeneous Linear Equations: Existence and Uniqueness of Solutions] Let A be an n × n matrix. Consider the following system of homogeneous
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Unformatted text preview: linear equations: Ax = 0 (4) (a) Suppose A is non-singular. 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 non-singular 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.

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Problem_Set_4 - linear equations Ax = 0(4(a Suppose A is...

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