problems4.1 - Math 307 Problems for section 4.1 1 For the following matrices nd(a all eigenvalues(b linearly independent eigenvectors for each

problems4.1 - Math 307 Problems for section 4.1 1 For the...

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Math 307: Problems for section 4.1 November 14, 2012 1. For the following matrices find (a) all eigenvalues (b) linearly independent eigenvectors for each eigenvalue (c) the algebraic and geometric multiplicity for each eigenvalue and state whether the matrix is diagonalizable. A = bracketleftbigg 3 7 2 - 2 bracketrightbigg (calculate by hand) B = 1 - 3 3 3 - 5 3 6 - 6 4 (calculate using Matlab/Octave or otherwise) C = 1 2 1 2 0 - 2 - 1 2 3 (calculate using Matlab/Octave or otherwise) 2. Find a 3 × 3 real, non-zero ( i.e. not all entries zero) matrix which has all three eigenvalues zero. 3. (a) By hand find a matrix with eigenvalues λ 1 = 1 and λ 2 = 2 and corresponding eigen- vectors v 1 = bracketleftbigg 1 2 bracketrightbigg v 2 = bracketleftbigg 2 1 bracketrightbigg (b) Using Matlab/Octave or otherwise, find a matrix with eigenvalues

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