hw8 - voltage V ) and one current source (with current I )....

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Math 152, Spring 2010 Assignment #8 Notes: Each question is worth 5 marks. Due in class: Monday, March 15 for MWF sections; Tuesday, March 16 for TTh sections. Solutions will be posted Tuesday, March 16 in the afternoon. No late assignments will be accepted. 1. Consider a random walk with n states (this number n will be known for a given case, but we are also interested in considering it for general n ). It has the following properties: If it is in states 1 to n - 1 (that is, any state but the last) is is equally likely to stay in that state or move to the state numbered one higher. If it is in the last state (number n ) then it remains there forever. (a) Write down the transition matrix P for the case of three states (that is, n = 3). (b) For the system above with three states, if it starts in state 1, what is the chance it will be in state 3 after 3 time steps? (c) Write lines of MATLAB commands that would generate the matrix P for general n . Hint: use a for loop. 2. Consider the circuit in the figure below. It has one voltage source (with
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Unformatted text preview: voltage V ) and one current source (with current I ). Solve the circuit with these symbolic values for the sources. Determine the 2 2 fundamental matrix F of the problem. That is, the matrix such that E J = F V I where E is the voltage across the current source and J is the current through the voltage source. 3. Consider the matrix A = 1 1 2 1 1 3 1 2 7 (a) Calculate the determinant of A (because the determinant of A is not zero, we know it is invertible). (b) Calculate the inverse of A . 4. Suppose A and B are invertible, square matrices. Give an example to show that A + B is not necessarily invertible. 5. Let T be the linear transformation from R 5 R 5 which sends a vector x = ( x 1 ,x 2 ,x 3 ,x 4 ,x 5 ) to the vector T ( x ) = ( x 1 + x 2 ,x 2 + x 3 ,x 3 + x 4 ,x 4 + x 5 ,x 5 + x 1 ). (a) Write the matrix A that corresponds to T . (b) Is A is invertible? Justify briey....
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hw8 - voltage V ) and one current source (with current I )....

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