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# ass5 - ASSIGNMENT 5 MATH 270 WINTER 2010 DUE 5PM APRIL 7...

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ASSIGNMENT 5 MATH 270, WINTER 2010. DUE: 5PM, APRIL 7, 2010. Assignments must have your name, student number, course number and TA’s name on the first page. Please submit them by placing them in the box marked “Assignments” outside the Maths Office (Burn- side Hall, 10th floor). Show all working and justifications! (1) The matrices A = 1 4 1 - 1 1 - 1 - 1 1 - 1 1 1 - 1 1 - 1 - 1 1 - 1 1 and B = 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 belong to which of the following classes: Defective? Factorisable as A = LU or B = LU ? Factorisable as A = QR or B = QR ? Invertible? Normal? Permutation? Projection? Self-adjoint? Skew-symmetric? Unitary? Give reasons in each case! [ Total: 10 marks ] (2) (a) Show that a unitary matrix U must have | det ( U ) | = 1. Give an example of a unitary matrix whose determinant is not 1. [ 2 marks ] (b) If a diagonalisable matrix A has | det ( A ) | = 1, must A be unitary? [ 1 mark ] (c) Show that if A is skew-symmetric, then A + I is invertible.

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ass5 - ASSIGNMENT 5 MATH 270 WINTER 2010 DUE 5PM APRIL 7...

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