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Unformatted text preview: Mathematics 250 Fall 2006 Selected Solutions Assignment #8 Proof Practice # 8 : Suppose that A is an invertible k × k matrix, and that B is another k × k matrix such that  B A  < 1  A 1  . Show that B is invertible, and that  B 1  ≤  A 1  1 B A  A 1  . Response: Following the suggestion in the problem statement, we write B = A ( I A 1 ( A B )). We recall the standard formula ( ST ) 1 = T 1 S 1 for the inverse of a product of invertible matrices. This formula shows how to compute the inverse of a product, and in particular, shows that the product of invertible matrices is also invertible. From our formula above for B , we see that this implies that if A and I A 1 ( A B ) = I Z are invertible, then B will also be invertible. We are given that A is invertible; thus it only remains to establish that I Z is invertible. From Proof Practice #7, we know that if Z is a matrix of norm less than one, then I Z is invertible. We can estimate that  Z  =  A 1 ( A B )  ≤  A 1   A B  <  A 1  1  A 1  = 1 . The first inequality follows from submultiplicativity, and the second, strict, inequality, follows by the given assumption on  B A  =  A B  . Thus, I Z is indeed invertible, and therefore so is B . Furthermore, Proof Practice #7 also gives us an estimate for  ( I Z )  . Applying this and again using submultiplicativity, we can say that  B 1  =  ( I Z ) 1 A 1  ≤  ( I Z ) 1   A 1  ≤ 1 1  Z   A 1  . ( Best ) Since  Z  ≤  A 1   A B  < 1, we see that 1  Z  ≥ 1  A 1   A B  = 1  B A   A 1  > 0. Taking reciprocals, we see that 1 1 Z  ≤ 1 1 A 1   A B  . Using this in the estimate (Best), we finally conclude that  B 1  ≤ 1 1  A 1   A B   A 1  =  A 1  1  B A   A 1  , as desired. AS8.1 : a) List the standard basis for Λ 3 ( R 5 ). b) Compute 1 1 1 ∧ 1 1 3 2 ∧ 1 1 2 . c) Find the Gram matrix of the three vectors of part b), and compute the volume of the parallelopiped spanned by them, and the areas of its faces....
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 Fall '06
 RogerHowe
 Math, Linear Algebra, Vectors, Mercator projection, Standard basis

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