Math 240 Fall 2013
Study problems for Exam1
continued
Do not use a calculator.
7. Match the slope fields with their differential equations.
(A)
(C)
(D)
dy
x
dx
1 0 1 2 3
#
8. Assume we are given Ax=b with augmented matrix A = 0 1 0 0 5
0 0 0 0 0
#
a) Is A in RREF? If not, use elementary row operations to find a row equivalent matrix that is.
b) what is the rank of A? c) What is the rank of A#?. d) Does the augmented matrix represent a
consistent system? e) How many variables are in the system? f) are there any free variables, if
so how many? g) Write the solution set.
1 0  2 0
9) Use elementary row operations to find the RREF of A# = 0 1 1 7 then write the
0 0 2 4
1.
dy
 sin x
dx
(B)
2.
dy
 x y
dx
3.
dy
 2 y
dx
4.
solution to the corresponding system .
10. If A and B are both n x n matrices, is the following correct? (If not, state the condition(s)
which makes it correct.)
(A + B)2= A2 - 2AB + B2
2 3
11. Given A, find the most general matrix B such that AB= 0. A = 

 0 0
12. Find a matrix B (or state it is impossible) such that AB = AT. (Use A from problem # 11)
13. Let A = diag(1,2,3,w) where w is an arbitrary real number. what is A5? What is A-1?
Explain.
14. A matrix A is called involutory if A2= I. Carefully prove that if A is involutory
then (I – A) (I + A) = 0
Math 240 Fall 2013
Study problems for Exam1
continued
Some solutions (not guaranteed to be correct!)
 3s / 2  3t / 2
# 8g {(3+2s-t, 5, s, t)} #9 {(5,5,2)} #10 no # 11 
s, t arbitrary real
t 
 s
numbers. 12. Impossible. 13. Diag (1,32,243, w5)