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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 2by2. 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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This note was uploaded on 11/13/2011 for the course MATH 67 taught by Professor Schilling during the Winter '07 term at UC Davis.
 Winter '07
 Schilling
 Linear Algebra, Algebra, Equations, Integrals, Matrices

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