Assignment 6

Assignment 6 - MATH 223 Linear Algebra Winter 2008...

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Unformatted text preview: MATH 223, Linear Algebra Winter, 2008 Assignment 6, due in class Wednesday, March 5, 2008 1. Suppose that A is an m × n matrix and B is an n × m matrix. [Hence, both AB and BA are defined and square, but not necessarily of the same size.] Show that tr ( AB ) = tr ( BA ). If C and D are square matrices of the same size which are similar, show that tr ( C ) = tr ( D ). [ tr ( C ) is the trace of C , that is, the sum of the entries on the diagonal.] 2. For each of the following matrices A j ( j = 1 , 2 , 3) over the real numbers, (a) find all the eigenvalues of A j ; (b) find a basis for each eigenspace; (c) find the minimal polynomial min A j and the characteristic polynomial χ A j ; (d) find — if possible — an invertible matrix P j such that P- 1 j A j P j is invertible. If this is not possible, explain why. (e) Find A 11 1 . (2 11 = 2048 and 3 11 = 177147.) A 1 = - 7- 5 5 4 2- 1- 6- 6 7 , A 2 = 3- 2 3 10- 3 5 5- 4 7 , A 3 = 6 4- 1 4 6- 1- 4...
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Assignment 6 - MATH 223 Linear Algebra Winter 2008...

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