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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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]
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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, dened 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, dened 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 satises: 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...
View Full Document

Ask a homework question - tutors are online