MATH 307
FALL 2002
1. Let
1
A=
1
EXTRA CREDIT
1
2
1
3
Find all matrices B for which AB is the 2 × 2 identity matrix. Is A nonsingular? Explain.
2. Assuming that v3 6= 0 in the matrix

0
A =  v3
−v2

v2
−v1 
0
−v3
0
v1
find the kernel of A.
3. What can the rank of a 2 × 3 matrix be? Give an example of a matrix having each possible
rank.
4. Let V be the two dimensional subspace of R4 spanned by [4, 4, 4, 1]T , [1, 1, 1, 1]T and let P
be the orthogonal projection onto V . Find an orthonormal basis of V and then obtain the
matrix of P .
5. Let

6
4 
~b = 


0
−3

2 2
1 4 
A=

0 1
2 −2


Solve Ax = ~b in the least square sense.
6. Let

1

A= 3
1
1
2
0

4
2
2
Using Cramer’s rule, calculate the (1, 3) entry of A−1 as a quotient of two determinants.
(Recall that the the third column of A−1 is the solution of A~x = ~e3 .)
7. Let v be any nonzero n × 1 matrix, and define A = vv T . Show that A is an n × n matrix
of rank 1.
8. Let
.75
A=
.25
.6
.4
Diagonalize A and then use the result to compute A50 . Give an exact answer, not a calculator
approximation. (Suggestion: use the formula in Example 7, page 142.)