### 3×3 determinant

```ENGG2013 Unit 9
3x3 Determinant
Feb, 2011.
Last time
• 22 determinant
• Compute the area of a parallelogram by
determinant
• A formula for 2x2 matrix inverse
kshum
ENGG2013
2
Today
• 33 determinant and its properties
• Using determinant, we can
– test whether three vectors lie on the same plane
– solve 33 linear system
– test whether the inverse of a 33 matrix exists
kshum
ENGG2013
3
Vector Notation
• We will use two different notations for a point
in the 3D space
(x,y,z)
z
z
y
y
x
kshum
x
ENGG2013
4
22 determinant
Notation for 22 determinant :
+
–
How to calculate: ad – bc
- bc
kshum
ENGG2013
5
33 determinant
Notation for 33 determinant :
Definition:
kshum
ENGG2013
6
Rule of Sarrus
+
+
+
–
–
–
Pierre Frédéric Sarrus (1798 – 1861)
kshum
ENGG2013
7
Volume of parallelepiped
• Geometric meaning
– The magnitude of 33 determinant is the volume
of a parallelepiped
z
y
x
kshum
ENGG2013
8
Co-planar  zero determinant
• Determinant = 0
 Volume = 0
 the three vectors lie on
the same plane
z
y
A collection of vectors
are said to be co-planar
if they lie on the same plane.
x
kshum
ENGG2013
9
Det of Diagonal matrix
• Volume of a rectangular box
c
b
a
kshum
ENGG2013
10
Transpose has the same determinant
+
+
+
–
–
–
Compare with
kshum
ENGG2013
11
Volume of parallelepiped
• In computing the volume of a
parallelepiped, it does not matter
whether we write the vector
horizontally or vertically in the
determinant
z
Volume of parallelepiped with vertices
(0,0,0), (1,2,0), (2,0,1), (–1, 1, 3) equals to
the absolute value of
y
or
x
kshum
ENGG2013
12
Question
• Do (1,1,1), (2,3,4), (5,6,7) and (8,9,10) lie on
the same plane?
kshum
ENGG2013
13
Cramer’s rule
• If the determinant of a 33 matrix A is non-zero, we can solve the linear
system A x = b by Cramer’s rule.
• The solution to
or equivalently
A
is
x
b
Gabriel Cramer (1704-1752)
kshum
ENGG2013
14
PROPERTIES OF DETERMINANT
kshum
ENGG2013
15
How to show that Cramer’s rule
• The Cramer’s rule is a theorem, which requires
a proof, or verification.
• We need some properties of determinant.
kshum
ENGG2013
16
Properties of determinant
1. Taking transpose does not change the value
of determinant
We have already verified this property in p.11
kshum
ENGG2013
17
Meta-property
• Because
1. After taking the transpose of a matrix, columns
become rows, and rows become column.
2. Taking the transpose of a matrix does not
change the value of its determinant.
• Therefore, any row property of determinant is
automatically a column property, and vice
versa.
kshum
ENGG2013
18
Properties of determinant
2. If any row or column is zero, then the
determinant is 0.
For example
kshum
ENGG2013
19
Properties of determinant
3. If any two columns (or rows) are the
identical, then the determinant is zero.
For example, if the second column and the third
column are the same, then
kshum
ENGG2013
20
Properties of Determinant
4. If we exchange of the two columns (or two
rows), the determinant is multiplied by –1.
For example, if we exchange the column 2 and
column 3, we have
The first kind
of elementary
row operation
kshum
ENGG2013
21
Multiply by a constant
5. If we multiply a row (or a column) by a
constant k, the value of determinant
increases by a factor of k.
The 2nd kind
of elementary
row operation
For example, if we multiply the third row by a
constant k,
kshum
ENGG2013
22
6. If a row (or column) of a determinant is the
sum of two rows (or columns), the
determinant can be split as the sum of two
determinants
For example, if the first column is the sum of two column vectors, then
we have
kshum
ENGG2013
23
Properties of Determinant
7. If we add a constant multiple of a row
(column) to the other row (column), the
determinant does not change.
The 3rd kind
of elementary
row operation
For example, if we replace the 3rd column by the
sum of the 3rd column and k times the 2nd
column,
kshum
ENGG2013
24
Summary on the effect of the
elementary row (or column)
operations on determinant
• Exchange two rows (or columns)  change
the sign of determinant
• Multiply a row (or a column) by a constant k
 multiply the determinant by k
• Add a constant multiple of a row (column) to
another row (or column)  no change
kshum
ENGG2013
25
Proof of the Cramer’s rule
• The solution to
is
Verification for x1: Substitute the value of b1, b2
and b3 in the first column of A.
Verification for x2: Substitute the value of b1, b2
and b3 in the second column of A.
Cramer’s rule in wikipedia
Etc.
kshum
ENGG2013
26
Because x1, x2, x3 satisfy the system of linear equations, we have
By substitution
Property 6
Property 5
=0
kshum
ENGG2013
=0
By Property 3
27
MINOR AND COFACTOR
kshum
ENGG2013
28
Another way to compute det
Group the six terms as
33 determinant can be computed in terms of 22 determinant
kshum
ENGG2013
29
Minor and cofactor
• A minor of a matrix is the determinant of
some smaller square matrix, obtained by
removing one or more of its rows and
columns.
• Notation: Given a matrix A, the minor
obtained by removing the i-th row and j-th
column is denoted by Aij. It is also called the
minor of the (i,j)-entry aij in A.
kshum
ENGG2013
30
Expansion on the first row
Minor of a1
kshum
ENGG2013
Minor of b1
Minor of c1
31
Expansion on the second row
Minor of a2
kshum
ENGG2013
Minor of b2
Minor of c2
32
Expansion on the third row
Minor of a3
kshum
ENGG2013
Minor of b3
Minor of c3
33
The sign pattern
Expansion
on the first row
Expansion
on the second row
Expansion
on the last row
kshum
ENGG2013
34
Column expansion
We have similar recursive formula for
determinant by column expansion
For example,
Computation on the third column is easy,
because there are lots of zeros.
kshum
ENGG2013
35
Cofactor
• The minor together with the appropriate  sign is called
cofactor.
• For
The sign
The cofactor of Cij is defined as
Expansion on the i-th row (i=1,2,3):
The minor of aij
Expansion on the j-th column (j=1,2,3):
kshum
ENGG2013
36
A formula for matrix inverse
Suppose that det A is nonzero.
(Beware of the
subscripts)
Usually called the adjoint of A
Three steps in computing above formula
1. for i,j = 1,2,3, replace each aij by cofactor Cij
2. Take the transpose of the resulting matrix.
3. divide by the determinant of A.
kshum
ENGG2013
37
A Quotation
Algebra is but written geometry;
geometry is but drawn algebra.
--- Sophie Germain (1776-1831)
L'algèbre n'est qu'une géométrie écrite; la géométrie n'est qu'une algèbre figurée
kshum
ENGG2013
38
```