G Problem 1. (4 points) Suppose A is a matrix of size 5 by 5. Which of the following statements must be TRUE? (I) Eigenvalue of A can be zero, but eigenvector of A cannot be a zero vector. (II) If λ is an eigenvalue of A, then the nullity of A − λI must be at least one. (III) If det(A − λI) = −(λ − 5)2 (λ + 4)3 then λ = 5 is said to have algebraic multiplicity 2 and λ = −4 is said to have algebraic multiplicity 3. (A) I only (B) II only (C) III only (D) II and III only (E) I, II, and III

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

69.64%

step-by-step explanation: