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
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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
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Vector Notation
• We will use two different notations for a point
in the 3D space
(x,y,z)
z
z
y
y
x
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22 determinant
Notation for 22 determinant :
+
–
How to calculate: ad – bc
- bc
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ad
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33 determinant
Notation for 33 determinant :
Definition:
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Rule of Sarrus
+
+
+
–
–
–
Pierre Frédéric Sarrus (1798 – 1861)
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Volume of parallelepiped
• Geometric meaning
– The magnitude of 33 determinant is the volume
of a parallelepiped
z
y
x
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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
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Det of Diagonal matrix
• Volume of a rectangular box
c
b
a
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Transpose has the same determinant
+
+
+
–
–
–
Compare with
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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
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Question
• Do (1,1,1), (2,3,4), (5,6,7) and (8,9,10) lie on
the same plane?
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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)
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PROPERTIES OF DETERMINANT
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How to show that Cramer’s rule
does give the correct answer?
• The Cramer’s rule is a theorem, which requires
a proof, or verification.
• We need some properties of determinant.
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Properties of determinant
1. Taking transpose does not change the value
of determinant
We have already verified this property in p.11
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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.
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Properties of determinant
2. If any row or column is zero, then the
determinant is 0.
For example
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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
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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
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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,
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An additive property
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
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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,
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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
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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.
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Because x1, x2, x3 satisfy the system of linear equations, we have
By substitution
Property 6
Property 5
=0
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=0
By Property 3
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MINOR AND COFACTOR
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Another way to compute det
Group the six terms as
33 determinant can be computed in terms of 22 determinant
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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.
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Expansion on the first row
Minor of a1
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Minor of b1
Minor of c1
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Expansion on the second row
Minor of a2
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Minor of b2
Minor of c2
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Expansion on the third row
Minor of a3
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Minor of b3
Minor of c3
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The sign pattern
Expansion
on the first row
Expansion
on the second row
Expansion
on the last row
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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.
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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):
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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.
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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
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