FINAL EXAM, M340L, F94 ā Nortonās section
Directions: Do all problems and show your work clearly. Put a box around your final answer for each
problem. You must show your work and justify your answers to receive credit. No books, notes, or calculators
are permitted, except one 8.5 x 11 sheet of notepaper with your own notes. Use the space provided; continue
on the back if you need more space.
There are ten questions, each worth 10 points; 100 points total.
1. Use row reduction to find all solutions of the following system. Express your answer in vector
parametric form.
2x+ 2z= 4
y+ 2z+w= 4
x+z+w= 4.
2. Let Wbe the subspace of R4spanned by the vectors (1,0,1,0),(2,1,ā1,0),and (1,2,ā5,0).
(a) Find a basis for W.
(b) What is the dimension of W?
(c) Let x= (6,2,0,0). Find [x]B, where Bis the basis found in part (a).
3. Let
A=ļ£®
ļ£°
10 20 30
6 12 24
12 15 1994
ļ£¹
ļ£».
(a) Find the determinant of A.
(b) Find the determinant of 2A.
4. Let
A=ļ£®
ļ£°
1 2 0
1 3 0
0 0 3
ļ£¹
ļ£».
(a) Find Aā1.
(b) Use your answer to (a) to solve Ax= (1,2,5) for x.
5. Let
A=ļ£®
ļ£°
1 0 1 0
2 1 ā1 0
1 2 ā5 0
ļ£¹
ļ£».
(a) Find bases for the row space, column space, and null space of A.
(b) Define rank. What is the rank of A?
6. Let Abe an nĆnmatrix. List at least ten different conditions that are all equivalent to the statement
āAis invertibleā. (Do not include this statement itself as one of the ten.)
(Scoring: 1 point each; 10 points maximum.)
7. Let
A=ī1 2
ā1 4 ī.
(a) Find the eigenvalues of A.
(b) For each eigenvalue, find a corresponding eigenvector.
(c) Find a diagonal matrix Dand an invertible matrix Psuch that A=PD P ā1. (Check your answer.)
8. Consider the system
1
