mela_exam2

# mela_exam2 - 5 Under what condition is A is diagonalizable...

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Math 107 (06) – Exam 2 – Spring 2009 Name: Problem 1. A tank initially contains 600 l of salted water whose salt concentration is 0.5 kg/l . Salted water whose salt concentration is 0.25 kg/l ﬂows into the tank at the rate of 12 l/min . The mixture ﬂows out at the same rate. Find the salt content after 20 min.

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Problem 2. Find the solution of the diﬀerential equation y ± ( x ) + 2 xy ( x ) = x , with initial condition y (1) = 2.
Problem 3. Suppose that p ( λ ) = λ 4 ( λ - 2)( λ + 1) 3 ( λ 2 + 16) 2 is the characteristique polynomial to a certain 12th order (homogeneous) linear diﬀerential equation with constant coeﬃcients. What is a fundamental set of solutions? Problem 4. Suppose that a 4 × 4-matrix A has the characteristique polynomial P ( λ ) = ( λ - 1) 2 ( λ - 3)( λ

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Unformatted text preview: + 5) . Under what condition is A is diagonalizable? Problem 5. Condisder L ( y ) = y ±±±± + 2 y ±±± + 2 y ±± . (a) Solve L ( y ) = 0. (b) Solve L ( y ) = cos( x ). Problem 6. Consider the following matrix A = 2-4 5 1 2-1 4 3 1 5 . (a) What is the determinant of A ? (b) What is the trace of A ? (c) What are the eigenvalues of A ? Problem 7. Consider the matrix A = 1 0 1 0 0 2 0 1 1 . Find the eigenvalues and eigenvectors of A . Is A diagonalizable? Justify your answer. . Problem 8. Let A and B be two invertible matrices. Show that AB and BA have the same characteristic polynomial....
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mela_exam2 - 5 Under what condition is A is diagonalizable...

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