Matrix Methods
Midterm 1: October 1, 2003
Remember to show your work. A correct answer without explanation receives no credit.
This exam is closed-book, closed notes, no calculators. You are allowed a one-page cheat sheet.
If you find that the arithmetic for a given problem seems complicated, go back and check your work. I
have written the exam so that the calculations are not hard.
Wherever possible, I recommend that you check your arithmetic by plugging in to the original equations.
1. Use the defined Aand bfor all parts of this problem.
(a) Find any conditions on bso that Ax=bhas a solution, for A=
1 2 0 3
0 0 0 0
2 4 0 7
and b=
b1
b2
b3
.
(b) Describe the nullspace of A. Give both a formula and a geometric description.
(c) Find the general solution to Ax=b, when the solution exists.
(d) Describe the column space of A. Give both a formula and a geometric description.
(e) What is the rank of A?
2. (a) Use elimination to find L,D, and Uso that A=LDU if A=
aaa
a b b
a b c
.
(b) Under what conditions on a,b,c, and dis Ainvertible?
3. In this problem, consider the vector space M2×2of 2 ×2 matrices, and use the same definition of Aand
Bin all parts of this problem.
(a) Describe a subspace of M2×2that contains A="1 0
0 0 #but not B="0 0
0−1#.
(b) If a subspace of M2×2contains Aand B, must it contain I?
4. Answer the following questions with complete explanations.
(a) Does the set of vectors of the form
a
a+ 2
a
form a vector subspace in R3?
(b) If Ais an n×ninvertible matrix, is (AT)2always invertible? If so, describe its inverse.
(c) Given that A,B, and Care n×nmatrices, suppose that Cis invertible, (A−B)C= 0 and B=AT.
Prove that Ais symmetric.
Extra Credit: Suppose Ais a 2 ×1 matrix and Bis a 1 ×2 matrix. Is it possible for C=AB to be
invertible? Explain why or why not.
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