Find the characteristic polynomial of the matrix, using either a cofactor expansion or the special formula for 3times3 determinants. [Note: Finding the characteristic polynomial of a 3times3 matrix is not easy to do with just row operations, because the variable lambda is involved.]

Among the most famous of all meteor showers are the perseids, which occur each year in early august. in some areas the frequency of visible perseids can be as high as 40 per hour. given that such sightings are poisson events, calculate the probability that an observer who has just seen a meteor will have to wait at least five minutes before seeing another.