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

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.

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=

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=, b=[-4 3]^t by

i) using the SVD of A.

ii) solving the system A^t Ax = A^tb