5.3soln - Solutions to problems from section 5.3 2. Let P =...

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Unformatted text preview: Solutions to problems from section 5.3 2. Let P = bracketleftbigg 2- 3- 3 5 bracketrightbigg , D = bracketleftbigg 1 0 1 / 2 bracketrightbigg , and A = PDP- 1 . Find A 4 . Solution: First, since det P = 1 we know that P- 1 = bracketleftbigg 5 3 3 2 bracketrightbigg so that A 4 = ( PDP- 1 ) 4 = PD 4 P- 1 = bracketleftbigg 2- 3- 3 5 bracketrightbiggbracketleftbigg 1 4 (1 / 2) 4 bracketrightbiggbracketleftbigg 5 3 3 2 bracketrightbigg = bracketleftbigg 2- 3- 3 5 bracketrightbiggbracketleftbigg 1 1 / 16 bracketrightbiggbracketleftbigg 5 3 3 2 bracketrightbigg = bracketleftbigg 2- 3- 3 5 bracketrightbiggbracketleftbigg 5 3 3 / 16 1 / 8 bracketrightbigg = bracketleftbigg 151 / 16 45 / 8- 225 / 16- 67 / 8 bracketrightbigg 6. Find the eigenvalues and a basis for each eigenspace of the matrix 4 0- 2 2 5 4 0 0 5 = - 2 0- 1 1 2 1 5 0 0 5 0 0 4 1 2 1 4- 1 0- 2 Solution: The eigenvalues are 5 and 4. - 2 1 , 1 is a basis for the 5-eigenspace and - 2 4 5 is a basis for the 4-eigenspace. 8. Diagonalize the matrix A = bracketleftbigg 5 1 5 bracketrightbigg if possible. Solution: Since A is upper triangular, its eigenvalues are the entries on the main diagonal....
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This note was uploaded on 01/28/2010 for the course MATH 307 taught by Professor Axenovich during the Fall '08 term at Iowa State.

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5.3soln - Solutions to problems from section 5.3 2. Let P =...

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