8 State whether each of the following statments is true or false In each case

8 state whether each of the following statments is

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Unformatted text preview: 8) State whether each of the following statments is true or false. In each case give a brief reason. a) The matrix 0 1 0 0 0 1 0 0 0 has no eigenvectors. b) The vector bracketleftbigg bracketrightbigg is an eigenvector of the matrix bracketleftbigg 0 1 1 0 bracketrightbigg . c) If λ is an eigenvalue of the matrix A , then λ 3 is a eigenvalue of the matrix A 3 . d) If 0 is an eigenvalue of the matrix A , then A is not invertible. 9) Find, if possible, a matrix A obeying A 3 = bracketleftbigg − 34 − 105 14 43 bracketrightbigg 10) Find a 3 × 3 matrix M having the eigenvalues 1 and 2, such that the eigenspace for λ = 1 is a line whose direction vector is [2 , , 1] and the eigenspace for λ = 2 is the plane x − 2 y + z = 0. 11) Consider a population which is divided into three types and reproduces as follows: 70% of the offspring of type 1 are type 1, 10% are type 2 and 20% are type 3 10% of the offspring of type 2 are type 1, 80% are type 2 and 10% are type 3 10% of the offspring of type 3 are type 1, 30% are type 2 and 60% are type 3 All three types reproduce at the same rate. Let x i ( n ) denote the fraction of generation n which is of type i , for i = 1 , 2 , 3. a) Find a matrix A such that vectorx ( n + 1) = Avectorx ( n ). b) Find the eigenvalues of A . c) Is there an equilibrium distribution, i.e. a vector vectorx such that vectorx ( n ) = vectorx for all n if vectorx (0) = vectorx ? If so, find it. 12) Consider the following mass-spring system on a frictionless plane. Both masses are 1 kg. and the natural x 1 x 2 k 1 m 1 k 2 m 2 length of both springs is 1 m. Their spring constants are k 1 and k 2 . Let x i , i = 1 , 2 denote the distance of mass i from the wall at the left. a) Use Newton’s law to find the equations of motion of the masses. b) Write these equations as a first order system of differential equations. March 31, 2011 Eigenvalues and Eigenvectors 26 13) The circuit below is known as a twin–T RC network and is used as a filter. Find a system of equations that determine the various currents flowing in the circuit, asssuming that the applied voltage, V = V ( t ), is given. V R 1 R 2 R 3 C 3 R L C 1 C 2 Solutions 1) Find the eigenvalues and eigenvectors of a) bracketleftbigg − 2 1 3 bracketrightbigg b) bracketleftbigg − 3 − 2 15 8 bracketrightbigg c) bracketleftbigg 3 2 − 1 1 bracketrightbigg Solution. a) Call the matrix A . The eigenvalues of this matrix are the solutions of det( A − λI ) = det bracketleftbigg − λ − 2 1 3 − λ bracketrightbigg = ( − λ )(3 − λ ) − ( − 2) = λ 2 − 3 λ + 2 = 0 or λ = 1 , 2 . The eigenvectors corresponding to λ = 1 are all nonzero solutions of the linear system of equations ( A − I ) vectory = bracketleftbigg − 1 − 2 1 2 bracketrightbiggbracketleftbigg y 1 y 2 bracketrightbigg = bracketleftbigg bracketrightbigg = ⇒ bracketleftbigg y 1 y 2 bracketrightbigg = c bracketleftbigg 2 − 1 bracketrightbigg , c negationslash = 0 and those corresponding to...
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