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quiz2_S2005

# quiz2_S2005 - singular(Hint It is helpful to notice that...

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Name Class Time MATH 202 - QUIZ # 2 Due Monday, April 18 at 2PM Covers 5.3, 5.4, Chapter 6, 7.1 and 7.2 of the text Time: 60 minutes Please show all work. All answers must be justiFed! Books, notes, calculators, are not permitted on this quiz. As part of your obligations under the Honor Code, do not discuss this quiz with anyone until after the Monday 2PM deadline. WRITE OUT AND SIGN THE PLEDGE: I pledge my honor that I have not violated the Honor Code during this examination.

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1. (8 points) Find the eigenvalues and a basis for the corresponding eigenspaces for A = 1 1 1 0 2 1 0 0 1
2. (10 points) Consider a discrete dynamical system ~x ( t + 1) = A~x ( t ) where the 2 × 2 matrix A has eigenvalues λ 1 = 1 / 3 and λ 2 = 5 / 4 with corresponding eigenvectors ~v 1 = " 2 3 # and ~v 2 = " 5 1 # (a) If ~x (0) = " 1 1 # then Fnd an explicit formula for ~x ( t ). (b) Describe the behavior of the system as t goes to inFnity.

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3. (8 points) Consider the matrix A = 3 x 1 1 1 x x 3 1 (a) Expand det A to get a polynomial in x . (b) A matrix is called singular if it is not invertible. Which choices of x will make the matrix A

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Unformatted text preview: singular? (Hint: It is helpful to notice that there is one obvious choice of x that would make the columns of A dependent.) 4. (12 points) Consider the vectors ~v 1 = 1 1 1 1 , ~v 2 = 1 1 , ~v 3 = 1-1 (a) Compute the volume of the 3-dimensional parallelogram in R 4 with edges ~v 1 , ~v 2 , ~v 3 . (b) Find the vector in V = span( ~v 1 , ~v 2 , ~v 3 ) that is closest to ~ b = 1 2 3 4 . 5. (12 points) True or False: As usual, justify your answers! (a) If A is a square matrix then A and A T have the same eigenvalues. (b) If det A = 1 then A is an orthogonal matrix. (c) If A is an orthogonal matrix, and a ij is any entry of the matrix A then-1 ≤ a ij ≤ 1. (d) Suppose that A and B are n × n matrices, that det A = 3 and that det B = 6. Then det((2 A ) T B-1 ) = 1....
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quiz2_S2005 - singular(Hint It is helpful to notice that...

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