(
b
) W
h
e
nC=A
5
,
det(C) =
.
(c)
If D is obtained from A by interchanging rows three and
five, then
det(D) =
.
(d)
If E is obtained from A by replacing row four with negative
two times row four, then
det(E) =
.
(e)
What can you say about the kernel of the linear
transformation T:
6
→
6
defined by T(
x
)=A
x
? Why??
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6. (10 pts.)
Obtain an LU factorization for the matrix A below.
1
2 1
4
A=
3

600
6 12 3 12
7. (10 pts.)
Suppose that the matrix A has the following LU
factorization:
100
3

14
8

710
0

721
5
32
5 1
001
4
.
(a) If
b
is an element of
3
, the problem of solving the equation
A
x
=
b
may be handled by solving a pair of matrix equations in a
particular order.
Give the equations, complete with their
coefficient matrices, in the correct order.
Use all of the
information at hand, but do not attempt to solve either equation.
(b)
Do the columns of A span
3
? How do you know??
(c)
Are the columns of A linearly independent? How do you know??
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 Spring '09
 JULIANEDWARDS
 Linear Algebra, linear transformation, Det

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