MA 237 Fall 2008
Lecture #31
October 29th
BRING TO CLASS:
This lesson plan.
Attendance Sheet
TODAY WE WILL:
Take attendance.
Section 4.3
Homework: #1 and #2: parts a,b,c,e,g,i
I.
Learn common mistake to avoid.
II.
See physical interpretation of the
determinant.
III.
Finding inverses.
OBJECTIVE:
First, spend some time explaining that you can add multiples of another row, but you
can’t add multiples of the row you are modifying.
The Area Under a Parallelogram
Geometrically, the determinant of a matrix corresponds to the area under a parallelogram
defined by the column of the matrix.
determinant of a 2x2 matrix  area of a parallelogram
determinant of a 3x3 matrix  volume of a parallelopiped
determinant of a 3x3 matrix  hyper-volume of a hyper-parallelopiped
Some examples:
L
0
1. Consider the rectangle defined by   and   .
0
H 
L 0 
Graph. Clearly the area is LH. The det of 
 is also LH.
0 H 
L
a
2. Consider the rectangle defined by   and   .
0
H 
L a 
Graph. With a little work, the area is LH. The det of 
 is also LH.
0 H 
L
0
3. Consider the rectangle defined by   and   .
b 
H 
L 0 
Graph. With a little work, the area is LH. The det of 
 is also LH.
b H 
5L 
0
4. Consider the rectangle defined by   and   .
0
H 
5L 0 
Graph. Clearly, the area is 5LH. The det of 
 is also 5LH.
 0 H
Why the determinant? It helps to keep in mind that the determinant can be proven (page
260) to be the only function satisfying these properties that we also expect for
parallograms:
1 0
(the square 
 has area 1)
0 1
(b) satisfies the row interchange, row scalar and row additive properties
(a) yields 1 from the identity …
Digression: Relationship with the cross product
The cross product is a binary operation on two vectors in R3 that results in another
vector which is perpendicular to the plane containing the two input vectors.
The magnitude of the cross product can be interpreted as the unsigned area of the
parallelogram having a and b as sides.
In 3 dimensions, the absolute value of the cross product is equal to the determinant of
[ab]:
where i,j,k are unit vectors.
In 2 dimensions, the analog to the cross product of u and v is
.
The cross product is more than the determinant (is a vector, not a scalar) and less (cannot
be defined for arbitrary square matrices.)
Back to topic: Relationship with the inverse.
The inverse can be calculated with a formula involving the determinant.
Formulae for 1x1 matrices
The inverse of a 1x1 matrix A = [a] is A-1 =
Since [1/a]* [a]=[1].
1
[1] .
det A
Formulae for 2x2 matrices
a b 
1  d  b
The inverse of a 2x2 matrix A = 
is A-1 =
.

det A  c a 
c d 
1  d  b a b  1 0
Since
*
=
det A  c a   c d  0 1
General Formula
Generally, the inverse of an invertible A can be calculated as:
The adjoint of A (denoted adj(A)) is the transpose of the signed matrix of minors.
Applying the formula e xamples
1. 2x2 matrix
a b 
Let A= 
.
c d 
a. The determinant of A is ad-bc.
 d
b. The signed matrix of minors is 
 b
 c
.
 a 
 d
c. The transpose of the signed matrix of minors is 
 c
1  d  b
d. Thus the inverse is
.
det A  c a 
 b
 a 
2. 3x3 matrix
 1 1 4 
Let A=  2
1  1 .
 2 0
3 
e. The determinant of A is 15.
 3  4 2
f. The signed matrix of minors is  3 11 2 .
 3 9 3
 3 3  3
g. The transpose of the signed matrix of minors is  4 11 9  .
 2 2 3 
1
1
 1
 

5
5
 3 3  3  5
 4 11 3 
1 


h. Thus the inverse is
 4 11 9  = 
15 
 15 15 5 
 2 2 3   2
2
1 
 15 15 5 
This can be obtained from Cramer’s rule, which we are learning in the next class. In
the book, the formula is written somewhat differently:
Theorem 2 on page 270
Convenient because it shows the formula for the inverse for any e index.
b) Cramer’s Rule
Can be used to:
1) find the solution to general linear systems
2) provide a formula for the inverse of a matrix
Suppose that A is an nxn invertible matrix. Then the solution to the system Ax=b is
given by,
where Ai is the matrix found by replacing the ith column of A with b.