Math 337 – Fall 2004 Midterm Examination 1
Instructions. Show your work. Calculators are not permitted. Scoring: 12 points per
problem, except problem 8 which is 16 points (parts of problems are equally weighted unless
noted otherwise). This examination has eight problems; problem 5 through 8 are on the
back of this page.
Problem 1. (12 points) Where possible perform the following matrix calculations:
(a)
1
2
3
+
1
0
−2
(b) [ 1 2 ] 1
−1(c)1
−1[ 1 2 ]
(d)1 2
3 4 +
1
0
0
1
(e)1 0
2 1 1 2
0 1 (f)
01000
10000
00001
00010
00100
11111
22222
33333
44444
55557
Problem 2. (12 points) For each part, determine whether or not the given vector xis a
linear combination of the vectors in the given set S. Explain your reasoning.
(a) x= (1,2,3), S={(1,0,0),(0,1,0),(0,0,1)}
(b) x= (0,0,0), S={(1,2,3),(4,5,6),(7,8,9)}
(c) x= (2,3), S={(1,2),(2,1),(3,2)}
Problem 3. (12 points) Showing your work, find all 2 ×2 matrices Athat satisfy
A0 1
1 0 =0 1
1 0 A.
Problem 4. (12 points; 3 points per part) (a) Which three matrices E21,E31 ,E32 put A
into upper triangular form U?
A=
101
110
111
and E32E31 E21A=U.
(b) Multiply those E’s to get one matrix Mthat does elimination: M A =U. (c) If
possible, find M−1. (d) If possible, give the LU factorization of A.
(CONTINUED ON BACK)
