REVIEW_EXAM1
Do all assigned hws, follow class notes, and practice following exercises:
SE1. Consider the following systems of equations:
x1 3 x2  1  s
x1  rx2  5
i) For what values of r and s is the system of linear equations inconsistent?
ii) For what of r and s does the system of linear equations have infinitely many solutions?
iii) For what of r and s does the system of linear equations have unique solution?
 2 1 1 
3


 
SE2. Let A= 3 0 2 and v= 1 . Compute the following


 


 1
a)
Av
b)
(Av)
T
SE3. Let
x  2y
z3
x  3y  z  5
3 x  8 y  4 z  17
a) Write the coefficient matrix.
b) Write the augented matrix form.
c) Solve the system using reduced row echelon form.
SE4. Find the rank and nullity of the following matrix
1 1 1 4 
1 2 4 2 


 2 0 4 1 




1   1  1  
2 
       
1 
 2 1  8  
'
SE5. Is u    is a vector in the span of S   , 
?






3 
1

2

1


     
 


1
1
5

1
 
       
If so, express it as a linear combination of the vectors in S .
SE6. Determine whether the following system is consistent, if so , find its solution:
x1  x2  3x3  3
2 x1  x2  3x3  0
SE7 Determine whether the given sets are linearly independent.
1   1 1  3  
1   1 2 
        
1 1 
      
 0 1
a) S1    1 , 0 , 1  b) S 2   ,  ,  ,   
     
1  2  1  
1  0  1  0  
      
0 1  0  3  
SE7. For the following pairs of matrices
i) Compute the products AB and BA.
ii) Compare the results to see that AB  BA ; that is, in general, matrix multiplication is not
commutative
iii) Can you give an example of a pair of matrices A, B such that AB=BA?
2  2
1


SE8. A) For A= 3  1 0 and B=


 1 1
6 
EA  B
b) Let A=
i)
1 2  2
3  1 0  , 0find an elementary matrix E such that


0 3
4 
a b 
c d  .


Suppose that
ad  bc  0 , and B=
1  d  b
. Show that AB  BA  I 2 , hence A
ad  bc  c a 
1
is invertible and B  A .
ii) Prove that the converse of (i): If A is invertible then
SE9 Consider the system of linear equations
x1  x2  2 x3  2
x1  2 x2  3
 x2  x3  1
a)
c)
Ax  b .
A1 .
Write the system as a matrix equation
b) Show that A is invertible, and find
Use
A1 to solve the system.
SE10. Theorem 2.2 (section 2.3)-page 125
SE11. Theorem 1.9 (section1.7)-page 81
ad  bc  0 .