MATH 203 PRACTICE
MIDTERM I
Part 1
Please work out each of the given problems without the use of a calculator. Credit will be based on the steps that you show towards the final answer. Show your work.
Problem 1
Let
- Find A-1
- Find x if
Problem 2
Let
- Find A2
and A3.
- Make a conjecture about Ak.
- Use induction to prove your conjecture
from part B.
Problem 3
Let
Find a 2x1 matrix v such that Av = 2v.
Part 2
Please work out each of the given problems. Credit will be based on the steps that you show towards the final answer. Show your work.
Problem 4
Answer the following true or false and
explain your reasoning.
A.
If A and B
are n x n matrices and AB
= 0, then either A
= 0 or B = 0.
B. If A is a nonsingular matrix with
and
is a solution, then
Problem 5
Show that if AT = A-1, then |det(A)| = 1.
Problem 6
In the city of Digraphville, there are four
food-processing plants: the apple plant, the beet plant, the carrot plant, and the
dairy plant. There are one-way
roads from the apple plant to the beet plant and to the dairy plant. There is also a one-way road from the beet plant to the
carrot plant. There are two-way
roads from the apple plant to the carrot plant, from the beet plant to the dairy
plant and from the carrot plant to the dairy plant.
A.
Sketch the digraph for this situation
B.
Write down the adjacency matrix
C.
Use the adjacency matrix to determine how many ways are there to drive
from the apple plant to the dairy plant using no more than four roads counted
with multiplicity.
Problem 7
Prove that if A,
B, and C are
n x n matrices, then
A(B + C)
= AB + AC
Problem 8
Prove that if
v
and
w are solutions to the matrix equation Ax
=
b and if r + s = 0, then
rv +
sw
is a solution to the homogeneous equation Ax
= 0.
Extra Credit:
Write
down one thing that your instructor can do to make the class better and one
thing that you want to remain the same in the class.
(Any constructive remark will be worth full credit.)