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hw7 - Math 260 Spring 2012 Jerry L Kazdan Problem Set 7 Due...

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Math 260, Spring 2012 Jerry L. Kazdan Problem Set 7 Due: In class Thursday, March 1. Late papers will be accepted until 1:00 PM Friday . Unless otherwise stated use the standard Euclidean norm . 1. Say you have a matrix A ( t ) = (( a ij ( t ))) whose elements a ij ( t ) depend on a parameter t R . We say that A ( t ) is differentiable at t 0 if the limit lim h 0 A ( t 0 + h ) A ( t 0 ) h exists. We then call this limit A ( t 0 ). It is easy to show that A ( t ) is differentiable if and only if all of the elements a ij ( t ) are differentiable. a) Say A ( t ) and B ( t ) are n × n matrices that are differentiable and let C ( t ) := A ( t ) B ( t ) Show that C ( t ) is differentiable and give a formula for C ( t ) in terms of A , A , B , and B b) If A ( t ) is differentiable and invertible, show that A 1 ( t ) is differentiable and give a formula for dA 1 ( t ) /dt in terms of A , A 1 , and A . [ Suggestion: Imitate the proof for a 1 × 1 matrix.] 2. In each of the following find u ( x, t ). a) ∂u ∂t = 0 with u ( x, 0) = e x sin 2 x .
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