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Area, Volume,
and the
Determinant
5
...
In particular, we have only been interested in whether the determinant
of a matrix is zero or non-zero
...
Area and the Determinant
Let i1
=
[��]
and
v
=
[�� l
In Chapter 1, we saw that we could construct a paralle
logram from these two vectors by making the vectors i1 and
having i1 +
v as adjacent sides and
v as the vertex of the parallelogram, opposite the origin, as in Figure 5
...
1
...
Ut UJ + V) X1
0
Figure 5
...
1
Parallelogram induced by i1 and v
...
4
...
This gives
Area(it, v)
=
=
=
=
Area of Square - Area 1 - Area 2 - Area 3 - Area 4 - Area 5 - Area 6
(u1 + V1)(u2 + v2) -
1
2
V1V2 - U2V1 -
1
2
u1u2 -
1
2
V1V2 - U2V1 -
1
2
U1U2
U1U2 + U1V2 + U2V1 + V1V2 - V1V2 - 2U2V1 - U1U2
U1V2 - U2V1
We immediately recognize this as the determinant of the matrix
[�� ��]
=
[it v]
...
However, this would be
slightly incorrect as we have made a hidden assumption in our calculation above
...
Area(a, V)
AS
A6
u2
0
Figure 5
...
2
EXERCISE 1
...
=
[UU1]2
Area of the parallelogram induced by
and v
=
[VVJ2]
a and v
...
(a) it=
(b) it
=
�
[
]
�l
[
[ [
� l -n
v
=
=
"
Solution: For (a), we have
Area (it, v)
=
l [-� ;JI
det
=
2(2) - 3(-2)
=
10
EXAMPLE 1
For ( b), we have
(continued)
l [ � �JI
Area(it,v) = det
-
= 1(-2)-1(1)= -3
Now suppose that the 2 x 2 matrix A is the standard matrix of a linear transforma
tion L: IR
...
2
...
Moreover, the volume of the image parallelogram is
l [
]I = ldet (A [a v])I
Area (Ail,Av) = det Ail Av
Hence, we get
l ( [
Area (Ait,Av)= det A il
v
])l =ldetAlldet [il v]l =ldetAIArea(il,v)
(5
...
The result is illustrated in Figure 5
...
3
...
4
...
EXAMPLE2
Let A =
i1 =
[ �]
[� �]
and v =
in two ways
...
Determine the image of
[-�]
under L and compute the area determined by the image vectors
EXAMPLE2
Solution: The image of each vector under L is
(continued)
Hence, the area determined by the image vectors is
Area (L(i1), L(v))
= l [� ;11 = - =
det
41
18
4
Or, using (5
...
2 JR
...
Determine the image of the standard basis vectors
in the
4
x1
e1 ez
and
direction
under S
and compute the area determined by the image vectors in two ways
...
The Determinant and Volume
JR
...
EXAMPLE 3
Solution: The volume determined by il, v, and w is
(continued)
H : -�]
Volume(ii, V, W) = det
= I
- 71 = 7
The volume determined by Ail, Av, and Aw is
l [
Volume(Ail,Av,Aw) = det Ail
Av
Aw
H � :J
JI
-
= det
Moreover, det A =
l
= I
-
3851
=
385
-55, so
Volume (Ail
...
In general, if v1,
...
Then-volume of the parallelotope is
and if A is the standard matrix of a linear mapping L
:
�n
�
�n, then
n-Volume (Av1,
...
, Vn)
PROBLEMS 5
...
(b) Determine the image of il and v under the lin
ear mapping with standard matrix
A=
[� n
-
(c) Compute the determinant of A
...
A2 Let A =
tion R
:
[� �]
�2
�
be the standard matrix of the reflec
�2 over the line x2 = x1
...
A3 (a) Compute the volume of the parallelepiped induced by ii=
lH [=n
jl =
and w =
m
(b) Compute the determinant ofA=
[! -� �]
...
w=
...
� , i12 =
0
...
3
, V,,} be vectors in IR
...
Prove that the
n-volume of the parallelotope induced by i11,
Vn
AS (a) Calculate the 4-volume of the 4-dimensional
1
0
parallelotope determined by i11 =
=
0
2·
5
(b) Calculate the 4-volume of the image of this
A4 Repeat Problem A3 with vectors a=
V=
1
1
3
• • •
,
is the same as the volume of the parallelotope in
duced by i11,
...
,
Homework Problems
Bl (a) Calculate the area of the parallelogram induced
by il =
[�]
and v =
[� ]
(c) What is the volume of the image of the paral
lelepiped of part (a) under the linear mapping
in JR
...
with standard matrixA?
(b) Determine the image of il and v under the lin
ear mapping with standard matrix
A=
[� n
(c) Compute the determinant ofA
...
B2 LetA =
H
[� � l
: JR
...
2
be the standard matrix of the shear
in the x1 direction by a factor of t
...
B3 (a) Compute the volume of the parallelepiped indured by U=
[-H [ H
V=
=
[H
[ � -� H
and W=
(b) Compute the determinant ofA=
[_il
u -r H
B4 Repeat Problem B3 with vectors i1 =
_
v =
Hl· nl·
w
andA=
=
BS (a) Calculate the 4-volume of the 4-dimensional
1
1
1
1
parallelotope determined by i11 =
v 2=
,
1
2
1
3
,
2
0
V3 =
, and V 4 =
...
, V,1} be vectors in lR
...
Prove
is half the volume of the parallelotope induced by
that the factor by which a volume is multiplied un
2i11, V2,
...
der the composite map M
n-volume of the parallelotope induced by v1,
...
•
o
L is I det BAI
...
In particular, try to in
evaluation of determinants
...
These review suggestions are
why it is true
...
1 and 5
...
They may
not cover every idea you need to master
...
Write down a 3 x 3 matrix A and
small groups may improve your efficiency
...
Be
especially careful about signs
...
1)
calculate A(cof A)T
...
3 )
4
How and why are determinants connected to
volumes? (Section 5
...
1
ES Suppose that A is a 5 x 5 matrix and det A
E2 By row reducing to upper-triangular form, evaluate
2
3
7
-8
20
-6 - 1 -9
det
3
8 21 -17
3
[�
E4 Determine all values of k such that the matrix
=
7
...
12
0
2
0
0
0
0
0
0
3
0
Evaluate det 0
0
0
0
1
...
=
E7 Determine x by using Cramer's Rule if
2
(b) If A =
2x1 + 3x + x3 = 1
2
X1 + Xz - X3
-1
0
+ 2x3 =
0
what is the volume of the
-4
parallelepiped induced by Ail, Av, and Aw?
=
-2x1
[� � _;],
1
E8 (a) What is the volume of the parallelepiped induced by U =
[J [-H
'
=
and W=
[!]1
Further Problems
These exercises are intended to be challenging
...
V3(a, b, c)= (c - a)(c - b)(b - a)
Fl Suppose that A is an n x n matrix with all row sums
a
a2
I
b
b2
3
a
3
b
1
1
c
c2
c3
d
d2
d
I!
equal to zero
...
)
j =l
Prove that detA= 0
...
Prove that detA= ± 1
...
By
3
using arguments similar to those in part (a) (and
without expanding the determinant), argue that
F3 Consider a triangle in the plane with side lengths
a, b, and c
...
By using trigonometry, show that
V4(a, b, c, d)= (d - a)(d - b)(d - c)V3(a, b, c)
F5 Suppose that A is a 4 x4 matrix partitioned into 2x2
blocks:
A=
c= b cosA + a cosB
Write similar equations for the other two sides
...
(b) Give an example to show that, in general,
b2 + c2 - a2
F4 (a) Let V3(a, b, c) = det
[fy]
detA * detA1 detA4 - detA2 detA3
2 bc
[� � ��]
...
Without ex-
c2
panding, argue that (a - b), (b - c), and (c - a)
are all factors of V3(a, b, c)
...