**Unformatted text preview: **1 1 ! 1 1 ! 1 2 " # $ $ % & ' ' ∈ M 3 ( R ) is diagonalizable (you do not need to find eigenvectors!). E. If A = a b c d ! " # $ % & ∈ M 2 ( R ) with bc > 0, show that A must be diagonalizable over R . F. If A ∈ M n ( F ) has a basis of n vectors, all of them eigenvectors for the value λ , show A = I n . G. If A ∈ M n ( R ) is diagonalizable over R , show that there is B ∈ M n ( R ) so that B 3 = A . H. If A ∈ M n ( R ) and p ( x ) = c n x n + ⋅ ⋅ ⋅ ⋅ c 1 x + c ∈ R [ x ], set p ( A ) = c n A n + ⋅ ⋅ ⋅ ⋅ + c 1 A + c I n . If is an eigenvalue for A , show that p ( ) is an eigenvalue for p ( A ). Hand In 9A) §5.4 #4; §5.5 #2; 4; 14; §5.6 #22; 38; 42; Ci); Cii); Ciii) 9B) §5.6 #42 (use hint above); A; D; G; H Due Friday October 28...

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