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hw1sol

# hw1sol - Winter 2011 Math 67 Linear Algebra Homework 1...

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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 0 on both sides yields 0 = a + b and 1 = - 3 a - 2 b . Hence 1 1 - 3 - 2 a b = 0 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 0 yields 0 = a + b and 1 = 3 ia - 3 ib . This gives the matrix equation 1 1 3 i - 3 i a b = 0 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 + d a ad - bc = 1 0 0 1 .

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