HW3_S2010 - X 1 : (In other words, show that the columns of...

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Problem Set 3 - Due Thursday 03/18/10 Econ 3200 - Introduction to Econometrics Spring 2010 Cornell University Prof. Molinari 1. Consider the matrices A and B A = 2 4 1 2 1 0 1 1 3 5 ; B = 1 3 2 1 1 1 ± : Compute the following: (a) AB and BA (b) A 0 and B 0 (c) A 0 B 0 and B 0 A 0 (d) A + B 0 and 1 2 A 0 2 B (e) ( BA ) 1 and ( A 0 B 0 ) 1 (f) rank ( A ) ; rank ( B ) ; and rank ( BA ) 2. Let X be an n ± ( k + 1) matrix with n > k + 1 and rank ( X ) = k + 1 (thus, ( X 0 X ) 1 P = X ( X 0 X ) 1 X 0 : Note that X is NOT square and that X 1 DOES NOT EXIST! (a) Prove that P is symmetric. (b) Prove that P is idempotent. (c) Prove that I P is symmetric where I is an n ± n identity matrix. (d) Prove that I P is idempotent. 3. Consider the matrices X 1 and X 2 X 1 = 2 4 1 1 1 1 1 0 3 5 ; X 2 = 2 4 0 2 2 0 1 1 3 5 : (a) Show that X 1 ( X 0 1 X 1 ) 1 X 0 1 = X 2 ( X 0 2 X 2 ) 1 X 0 2 : 1
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(b) Compute the trace of X 1 ( X 0 1 X 1 ) 1 X 0 1 and compute the trace of X 2 ( X 0 2 X 2 ) 1 X 0 2 : (c) Show that the columns of X 2 are linear combinations of the columns of
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Unformatted text preview: X 1 : (In other words, show that the columns of X 2 span the same space as the columns of X 1 ). 4. Let X be an n & ( k +1) matrix with rank k +1 (thus, ( X X ) & 1 exists). Suppose Z = XG where G is a ( k + 1) & ( k + 1) matrix and G & 1 exists. Note that Z is another n & ( k + 1) matrix. Also note that X & 1 and Z & 1 DO NOT EXIST because they are not square matrices! (a) Prove that X ( X X ) & 1 X = Z ( Z Z ) & 1 Z : (b) Compute the trace of X ( X X ) & 1 X : 2...
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This note was uploaded on 04/14/2010 for the course ECON 3200 taught by Professor Neilsen during the Spring '08 term at Cornell University (Engineering School).

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HW3_S2010 - X 1 : (In other words, show that the columns of...

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