Determinants: summary of main results
ä A determinant of an n × n matrix is a real number associated
with this matrix. Its definition is complex for the general case → We
start with n = 2 then list important properties for this case.
DETERMINANTS
• Determinant of a 2 × 2 matrix is:
a b
a b det c d = ad − bc
• Notation : det (A) or c d
CHAP. 3
ä Next we list the main properties of determinants. Properties also
true for n × n case
ä Can be defined from GE. Det. = product of pivots in GE when
permutation is not used. Adjust signs when permuting. More later
Text: 3.1-3 – DET
10-2
ä Properties written for columns (easier to write) but are also true
for rows
Notation: We let A = [u, v] columns u, and v are in R2.
1
If v = αu then det (A) = 0.
ä det (A) = linear function of each column (individually)
ä det (A) = linear function of each row (individually)
- What is the determinant det [u, v + αu]?
4
ä Determinant of linearly dependent vectors is zero
Determinant of transpose
det (A) = det (AT )
ä If any one column is zero then determinant is zero
2
5
Interchanging columns or rows:
det (I) = 1
det [v, u] = −det [u, v]
3
6
Linearity:
Text: 3.1-3 – DET
Determinant of a diagonal:
det (D) = d1d2 · · · dn
det [u, αv + βw] = αdet [u, v] + βdet [u, w]
10-3
Determinant of Identity
10-4
Text: 3.1-3 – DET
7
Determinant of a triangular matrix (upper or lower)
Determinants – general definition
det (T ) = a11a22 · · · ann
8
Determinant of product of matrices [IMPORTANT]

a11 a12
 a21 a22
A=
..
 ..
an1 an2
Consider now the general
situation of n×n matrices:
det (AB) = det (A)det (B)
9
General idea: det of A is the sum of all possible products of one
entry per row of A. Each product has a sign.
Consequence: Determinant of inverse
det (A−1) =

· · · a1n
· · · a2n 

. . . .. 
· · · ann
1
ä We need to use permutations to define determinants
det (A)
- What is the determinant of αA?
What can you say about the determinant of a matrix which
satisfies A2 = I?
-
ä p = {i1, i2, · · · , in} is a permutation of {1, 2, · · · , n} if
each if the numbers 1, 2, ..., n is represented once and only once in
the list p
- Is it true that det (A + B) = det (A) + det (B)?
Text: 3.1-3 – DET
10-5
Example:
{2, 3, 1, 5, 4} is a permutation of {1, 2, · · · , 5}.
Any permutation is the result of a sequence of interchanges (or
swaps) applied to {1, 2, · · · , n} in which two keys are exchanged
each time.
ä For the above example: start with {1, 2, · · · , 5}, then swap
keys in positions 4 and 5 (result {1 2 3 5 4 }) then those in positions
1 and 2 (result {2 1 3 5 4 }) and finally keys in positions 2 and 3 to
get desired the result {2 3 1 5 4 }.
ä Consider now the case n = 3.
ä We will need to use all permutations of {1, 2, 3}.
ä Denote by σ the permutations
ä Here are all permutations of {1, 2, 3} with their signatures
{
{
{
{
{
{
• The signature of a permutation is (−1)np where np is the
number of swaps needed to rearrange {1, 2, · · · , n} into p.
10-7
Text: 3.1-3 – DET
Sign.
σ
- Obtain p = {3, 1, 4, 2} from {1, 2, 3, 4}
ä In the above example the signature is −1 since 3 swaps were
needed.
Text: 3.1-3 – DET
10-6
1
1
2
2
3
3
2
3
3
1
1
2
3
2
1
3
2
1
}
}
}
}
}
}
+1
−1
+1
−1
+1
−1
ä We will denote by sig(σ) the signature of σ
10-8
Text: 3.1-3 – DET
General definition of determinants (‘Big formula definition’)
X
det (A) =
sig(σ)a1σ(1)a2σ(2) · · · anσ(n)
σ
Where the sum runs over all (n!) possible permutations σ of
{1, 2 , · · · , n}.
Case n = 3
σ
{
{
{
{
{
{
1
1
2
2
3
3
2
3
3
1
1
2
3
2
1
3
2
1
}
}
}
}
}
}
Sign.
Det = Sum of:
+1
−1
+1
−1
+1
−1
+a11a22a33
−a11a23a32
+a12a23a31
−a12a21a33
+a13a21a32
−a13a22a31
- Let B be the matrix obtained from a matrix A by multiplying
What is the computational cost of evaluating the determinant
using the ’big formula definition’ ? [Hint: It is big!]
-
Text: 3.1-3 – DET
Text: 3.1-3 – DET
10-10
ä This gives a second definition of determinants – a recursive one.
Cofactors
ä Let Aij be the (n − 1) × (n − 1) matrix obtained from A
by deleting its i-th row and its j-th column.
ä We can expand det (A) with
respect to i-th row as follows:
Suppose columns 1 and 2 are swapped. Use the ‘big formula
definition’ to show that the determinant changes signs.
-
a certain row (or column) of A by a scalar α. Use the ‘big formula
definition’ to show that: det (B) = αdet (A).
10-9
ä Define C the matrix of cofactors, having entries:
Computing determinants from defining formula
−1
2 0 - Compute the following de
2 −1 3 terminant using the ‘Big formula
−1
0 2 definition’
cij = (−1)i+j det Aij
det (A) =
Pn
j=1
ä We know how do define determinants for n = 2. For n > 2
define determinant by expanding with respect to the first row.
Recursive definition of determinants: For n >2, the determinant of
a matrix A = [aij ] is the sum of the n terms a1j c1j , i.e.,
det (A) = a11c11 + a12c12 + · · · + a1nc1n
n
X
=
(−1)1+j a1j det (A1j )
aij cij
j=1
ä Note i is fixed. Can be done for any i [same result each time]
- Establish a recurrence relation that gives the cost of computing
det (A) using co-factors. Show that the cost is ≈ 2(n!)
ä Similar expressions for expanding column-wise
- Prove the above result for n = 3 [Hint: list permutations in a
certain order]
10-11
Text: 3.1-3 – DET
10-12
Text: 3.1-3 – DET
−1
2
−1
- Compute the following deter-
minant by using co-factors. Expand with respect to 1st row.
2
−1
0
0
3
2
Compute the above determinant by using co-factors. Expand
with respect to last row. Then expand with respect to last column.
1 −1
1
- Compute F2, F3, F4
1
−1
.
.
.
when Fn is the n-th di..
.. ..
Fn = mensional determinant:
1
1
−1
1 −1 -
Cramer’s rule
Notation: For any n × n matrix A and any b in Rn let Ai(b)
be the matrix obtained from A by remplacing its i-th column by b:
Ai(b) = [a1, a2, · · · , ai−1, b, ai+1, · · · , an]
Cramer’s rule Let A be an invertible n × n matrix and b in Rn.
The unique solution of the system Ax = b has entries given by:
xi = det (Ai(b))
det (A)
ä In addition the following formula for the inverse holds:
(continuation) Challenge: Show a recurrence relation between
Fn, Fn−1 and Fn−2. Do you recognize this relation? Compute the
first 8 values of Fn
-
A−1 =
1
det (A)
CT
where C be the matrix of cofactors.
Text: 3.1-3 – DET
10-13
-

2

Find the inverse of A = 1
3
1
−1
1

1
4 
−2
Areas and volumes
- Determine how x1(α) depends on α when x1(α) is the first
component of the solution of the system Ax = b, where

2

A= 1
3
10-15
1
−1
1

1
4 
−2
Text: 3.1-3 – DET
10-14
 
−2

b= α
1
ä Area of a parallelogram in R2 spanned by
points (0, 0), (a, b), (c, d), (a + c, b + d)
is:
2
ä Area of triangle in R spanned by the 1
det
points (x1, y1), (x2, y2), (x3, y3) is:
2
ä
Volume of a parallelogram in
3
R spanned by points (x1, y1, z1),
(x2, y2, z2), (x3, y3, z3) is
Text: 3.1-3 – DET
10-16
a
b
det
c d 

1 x1 y1 1 x2 y2
1 x3 y3 

x
y
z
1 1 1 det x2 y2 z2
x3 y3 z3 Text: 3.1-3 – DET
How to compute determinants in practice?
ä Co-factor expansion?? *Not practical*. Instead:
ä Perform an LU factorization of A with pivoting.
ä If a zero column is encountered LU fails but det(A) = 0
ä If not get det = product of diagonal entries multiplied by a sign
±1 depending on how many times we interchanged rows.
- Compute the determinants of the matrices


2 4
6
9 
A = 1 5
1 0 −12
10-17


0 −1 1
2
 1 −2 −1 1 

B=
 2
0
2
0 
−1 1 −1 −1
Text: 3.1-3 – DET