For each of the following statements, prove that it is true or give an example to show it is false. throughout, a e gm x r n unless otherwise indicated, and "ew" stands for eigenvalue. (this comes from the german "eigenwert." the corresponding abbreviation for eigenvector is "ev," from "eigenvektor." )

( a ) i f x i s a n e w o f a a n d p ~ c , t h e n x - p i s a n e w o f a - p i .
(b) if a is real and x is an ew of a, then so is -a.
(c) if a is real and x is an ew of a, then so is 5; .
(d) if x is an ew of a and a is nonsingular, then a-i is an ew of a-'.
(e) if all the ew's of a are zero, then a = 0.
(f) if a is hermitian and x is an ew of a, then ixi is a singular value of a.
(g) if a is diagonalizable and all its ew's are equal, then a is diagonal

Steve is scuba diving near his home in maui. at one point he is 100 feet below the surface. represent this number with a negative number. if he descends another 5 feet, what negative number will represents his.

Find the solution of each of the following.

Asystem of linear equations with more equations than unknowns is sometimes called an overdetermined system. can such a system be consistent? illustrate your answer with a specific system of three equations in two unknowns. choose the correct answer below. a. yes, overdetermined systems can be consistent. for example, the system of equations below is consistent because it has the solution nothing. (type an ordered pair.) x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 6 b. no, overdetermined systems cannot be consistent because there are fewer free variables than equations. for example, the system of equations below has no solution. x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 12 c. yes, overdetermined systems can be consistent. for example, the system of equations below is consistent because it has the solution nothing. (type an ordered pair.) x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 8 d. no, overdetermined systems cannot be consistent because there are no free variables. for example, the system of equations below has no solution. x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 24