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problem1) a) Let M and T be a metro city and a nearby district town, respectively. Our government is trying to develop infrastructure in T so that people shift to T. Each year 10% of T’s population moves to M and 1% of M’s population moves to T. What is the long term effect of this on the populations of M and T? Are they likely to stabilise?

b) The circuit in Figure 1 is given by the matrix equation

2466_Matrix equation.jpg

where v1(t) and v2 (t) are the voltages across the capacitors C1 and C2 at time t. Suppose the resistor R1 is 1 ohm, R2 is 2 ohms, C1 is 1 farad, C2 is 0.5 farad, there is an initial charge of 5 volts on C1 and 4 volts on C2 . Then solve the equation.

291_Circuit.jpg

c) Give an ex each to show that the following statements are false.

i) If A1 and A2 are n × n matrices which are similar to B1  and B2 , respectively, then A1 A2 is similar to  B1 B2 .

ii) Two n n ×matrices with the same characteristic polynomial mu  st be similar.

problem2) Find a unitary matrix U such that * U AU is upper triangular, where A=149_Matrix_5.jpg

Hence obtain an orthonormal basis for the linear operator T on R3 with respect to which the matrix of T is upper triangular. Here T is defined by A w.r.t. the standard orthonormal basis on R3.

problem3) Find the least squares solution to Ax = b, where A=1742_Matrix_6.jpg, b=[-4 3]t by

i) using the SVD of A.

ii) solving the system At Ax = Atb

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