Problem_Set_3

# Problem_Set_3 - (a Let A be an m × 1 matrix and let B be a...

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T.Mitra, Fall 2008 Economics 6170 Problem Set 3 (For practice only: do not hand in solutions) 1. [Matrix Multiplication] Suppose A is an m × n matrix, B is an n × r matrix, and C is an r × s matrix. Verify that: A ( BC ) = ( AB ) C 2. [Transpose of a Matrix] Suppose A is an m × n matrix, B is an n × r matrix. Verify that: ( AB ) 0 = B 0 A 0 3. [Rank of a Matrix] Let A be an m × n matrix, and suppose the n column vectors of A are linearly independent. (a) Show that m n. (b) Show that the row rank of A n. [Do not use the Rank Theorem to answer any of the parts of the problem. You can, of course, use any result, which appears in the lecture notes before the statement of the Rank Theorem]. 4. [Singular and Non-Singular Matrices]
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Unformatted text preview: (a) Let A be an m × 1 matrix and let B be a 1 × m matrix, where m ≥ 2 . Let C be the m × m matrix, deﬁned by C = AB. Show that C must be a singular matrix. (b) Let A be an m × n matrix and let B be an n × m matrix. Let C be the m × m matrix, deﬁned by C = AB. If n < m, can C be non-singular ? Explain your answer carefully. 5. [Inverse of a Matrix] Let A be an n × n matrix, which satisﬁes: a ij = ± 1 for all i,j ∈ { 1 ,...,n } with j ≤ i 0 otherwise Show that A has an inverse.[Do not use your computer to obtain the inverse matrix]. 1...
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