Signature Printed Name
Math 313 Final Exam Jerry L. Kazdan
April 29, 2005 1:30 — 3:30
Directions This exam has 12 problems (10 points each). Closed book, no calculators – but you
may use one 3”×5” card with notes.
1. Let Abe a 3 ×3 matrix with eigenvalues λ1, λ2,λ3and corresponding
(independent) eigenvectors V1,V2,V3which we can therefore use as a basis
(of course AVj=λjVj).
If X=aV1+bV2+cV3, compute AX,A2X, and A35Xin terms of λ1,λ2,
λ3,V1,V2,V3,a,band c(only).
2. Let A:= 1 4 11 −4
−1−2−5 6
0 4 12 5
−1 2 7 4 ,X0:= 1
0
0
1.Y:= −3
5
5
3, and Z:= 1
−3
1
0.
You are given that the vector X0is a particular solution of AX =Yand
Zis in the nullspace of A.
a) Find another solution (other than X0) of AX =Y.
b) If Zis a basis for the nullspace of A, find the general solution of AX =Y.
3. Let Abe an n×nmatrix of real numbers. Circle each of the following
statements that are NOT equivalent to: “the matrix Ais invertible”? [No
justification is needed.]
a) The columns of Aare linearly independent.
b) The columns of Aspan Rn.
c) The only solution of the homogeneous equations Ax = 0 is x= 0.
d) The linear transformation A:Rn→Rndefined by Ais 1-1.
e) The linear transformation A:Rn→Rndefined by Ais onto.
f) The rank of Ais n.
g) The transpose, AT, is invertible.
Score
1
2
3
4
5
6
7
8
9
10
11
12
Total
4. Let A,B, and Cbe n×ninvertible matrices.
a) Solve the equation C−1(2I+AM )C=Bfor the matrix M.
b) If 2 is not an eigenvalue of B, show that Mis invertible,
