Preparing for the Exam (2)

Here are a collection of relatively easy problems:

1) Can you find a 4 by 4 matrix, that is NOT upper-triangular with charpoly being

2) When you have a matrix, always ask yourself the following questions:
(a) What is the rank?
(b) Can I find the rank without computing?
(c) Are there any obvious conditions that would help finding out the rank without actual computation?
(d) What is the charpoly? As always, look for short cuts. For example, there might be obvious eigenvalues. (just like D, the differential transformation on the space of polynomials)

Here are some general strategy in solving problems.
-If you cannot solve the stated problem, try to solve a simpler problem, then slowly complicate the situations. For example, if the question involves a n by n matrix, try to consider the n = 2 case first. Completely solve the n = 2 case, and see if you get any hints from this special case to solve the general case. Nevertheless, solving n = 2 does not mean you have finished the problem, you must then go back to solve the general n case.