Are the following statements true or false? False 1. Finding an eigenvector of might be difficult, but checking whether a given vector is in fact an eigenvector is easy. False 2. A matrix is singular if and only if 0 is an eigenvalue of . False 3. If for some vector , then is an eigenvalue of . True 4. To find the eigenvalues of , reduce to echelon form. True 5. A number is an eigenvalue of if and only if the equation has a nontrivial solution .

Answer these questions showing your work plz

What is the sum of these mixed numbers 2 1/8 + 4 3/8

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.