1. No category

SM212 Sample Test 3
Multiple Choice
1. Let
,
,
. Which of the following matrix products is
not possible? (If it is possible, what is the result?)
a. AB
b. BC
c. CA
d. AA
e. AC
2. If A is an n x n matrix, which of the following statements is not equivalent to
the others?
a. A row reduces to the identity matrix (i.e. rref(A)=In).
b. A-1 exists.
c. det(A)=0.
d. AX=0 has only the trivial solution X=0.
e. They are all equivalent.
3. If
where y equals:
a. 24
b. 0
c. 5
d. 6
e. 39
, then the matrix equations
has a solution
,
4. The matrix
a.
b.
c.
d.
e.
has eigenvalues -1 and 4, if x equals:
0
1
2
3
4
5. The vectors
and
the general solution of the system
are eigenvectors of
. Then
is:
a.
b.
c.
d.
6. Use two steps of Euler’s method to approximate y(0.2), where y(x) is the
solution of the initial-value problem:
y’’-4y’+4y=0, y(0)=-2, y’(0)=1
a.
b.
c.
d.
0.64
-1.68
-1.78
-1.9
Long Response
7. Consider the system of differential equations (where x and y are function of t):
a.
b.
c.
d.
Rewrite the given systems of matrix normal form X’(t)=AX(t), X(0)=X0.
Solve the system IVP using eigenvalues and eigenvectors.
Solve the IVP system using Laplace transforms.
Use Euler’s method to estimate the x(.2) and y(.2) (Let Use Euler’s method
to estimate the x(.2) and y(.2) (Let
).
8. Consider the matrix
. Find the general solution of the system
X’=AX.
9. Consider the electrical network shown below. This network
is modeled by the following system of differential
equations:
Use the Laplace transform to find q(t) and i(t) given that
q(0)=1 C and i(0)=0 A.
10.Solve the given the system of differential equations using Laplace transforms:
11.Use Euler’s method to estimate the x(.2) and y(.2) (Let
of equations above.
) in the system
12. Consider the IVP y’’+3y’-4y=t+1, y(0)=1, y’(0)=0.
a. Solve for y analytically. What is y(.1)?
b. Write the second order IVP is a system of 1st order IVPs.
c. Use Euler’s method to estimate y(.1) using
.
d. What is the answer in part c not the same as the answer in part a?