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hw01 - exp A to be exp A = ∞ ± k =0 A k k A Show that...

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STA 6246 Linear Models Fall 2011 Homework #1 Due Friday September 9 1 Suppose A and B are n × n matrices. Prove from first principles, i.e. using the definition of a nonsingular matrix (see page 19 of the notes), that if either AB or BA is nonsingular, then both A and B are nonsingular. 2 Suppose A and B are n × n matrices. Recall that we use r ( M ) to denote the rank of the matrix M , i.e. dim( C ( M )) . Show that r ( AB ) min( r ( A ) , r ( B )) . 3 For a square matrix A , we “define”
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Unformatted text preview: exp( A ) to be exp( A ) = ∞ ± k =0 A k k ! . A Show that the series converges absolutely, so that this is a proper de±nition. B Find exp( A ) where A is given by A = 1 1 2 1 4 1 2 1 1 2 1 4 1 2 1 . 4 Suppose A and B are n × n matrices. Show from ±rst principles that if B is nonsingular, then r ( BA ) = r ( AB ) = r ( A ) ....
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