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Unformatted text preview: Winter 2011 Math 67 Linear Algebra Homework 1 Problem 1. Write down matrix equations of the form Ax = b that come from the integrals Z dx x 2- 5 x + 6 Z dx x 2 + 9 . The second matrix will have complex (imaginary) numbers in it. Both matrices will be 2-by-2. Solution 1. For the first integral we consider the equation 1 x 2- 5 x + 6 = a x- 2 + b x- 3 . Clearing denominators gives 1 = a ( x- 3) + b ( x- 2) and equating coefficients of x 1 and x on both sides yields 0 = a + b and 1 =- 3 a- 2 b . Hence 1 1- 3- 2 a b = 1 . Doing the same thing for the equation 1 / ( x 2 + 9) = a/ ( x- 3 i ) + b/ ( x + 3 i ) gives 1 = a ( x + 3 i ) + b ( x- 3 i ) . Equating coefficients of x 1 and x yields 0 = a + b and 1 = 3 ia- 3 ib . This gives the matrix equation 1 1 3 i- 3 i a b = 1 . Problem 2. If A = a b c d and A- 1 = 1 ad- bc d- b- c a then verify that AA- 1 = A- 1 A = 1 0 0 1 provided, of course, that ad- bc 6 = 0. Solution 2. To do this simply take the dot product of row i of A with column j of (the potential inverse) A- 1 . That is a b c d . d ad- bc- b ad- bc- c ad- bc a ad- bc = a d ad- bc- b c ad- bc a- b ad- bc- b a ad- bc c d ad- bc + d- c ad- bc c- b ad- bc +...
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