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Course 111: Algebra, 20th April 2007 To be handed in at tutorials on April 23rd and 24th. 1. Consider the matrix 1 0 1 A = 2 −2 −1 0 0 1 Write A in its Jordan form, J by determining the matrix V which transforms A according to J = V −1 AV . Use the formula in your notes to determine e2J . 2. Consider the matrix equation, Ax = b with A= 1 1 1 1+ ! , 0 < << 1 (1) Determine the condition number of the matrix A with respect to the ∞ norm. Is the matrix well or poorly conditioned? Given 2 2 b= ! , determine a solution, x to the matrix equation Ax = b. Determine what happens to the solution when A= 1 1 1 1 ! , 0 < << 1 and b is defined as above. Also, determine the effect of a small change in b on the solution by considering solutions when b = (2, 2)T and b = (2 + , 2)T and A is as given in Eqn 1.