Download Determinants - Computer Sciences - Lecture Slides and more Slides Operating Systems in PDF only on Docsity!
DETERMINANTS CHAP. 3
Determinants: summary of main results
ä We begin with the determinant of a 2 × 2 matrix
det
[
a b
c d
]
= ad − bc
Notation : det (A) or
a b
c d
ä Next we list the main properties of determinants.
ä Properties also true for n×n case. In fact they motivate
the definition of det (A) for n > 2.
ä Det. of an n × n matrix = product of pivots when
permutation is not used. Adjust signs when permuting.
I-
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 R 2 .
1 If v = αu then det (A) = 0.
ä Determinant of linearly dependent vectors is zero
ä If any one column is zero then determinant is zero
2 Interchanging columns or rows:
det [v, u] = −det [u, v]
3 Linearity:
det [u, αv + βw] = αdet [u, v] + βdet [u, w]
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 Determinant of transpose
det (A) = det (A T )
5 Determinant of Identity
det (I) = 1
6 Determinant of a diagonal:
det (D) = d 1 d 2 · · · dn
7 Determinant of a triangular matrix (upper or lower)
det (T ) = a 11 a 22 · · · ann
8 Determinant of product of matrices [IMPORTANT]
det (AB) = det (A)det (B)
9 Consequence: Determinant of inverse
det (A
− 1 ) =
1 det (A)
- What is the determinant of αA?
- What can you say about the determinant of a matrix
which satisfies A 2 = I?
- Is it true that det (A + B) = det (A) + det (B)?
I-
Determinants – general definition
ä Consider now the general situation of n × n matrices:
A =
a 11 a 12 · · · a 1 n
a 21 a 22 · · · a 2 n .. .
an 1 an 2 · · · ann
General idea: det of A is the sum of all possible products
of one entry per row of A. Each product has a sign.
ä Consider the situation n = 3.
ä We will need to use all permutations of [1, 2 , 3]
ä σ = [2, 1 , 3] is one such permutation. Its signature is
− 1 because we need one interchange to go from [1, 2 , 3]
to [2, 1 , 3] (swapped 1 and 2)
I-
ä σ = [3, 1 , 2] is another such permutation. It signature
is +1 because we need two interchanges to go from [1, 2 , 3]
to [3, 1 , 2]
ä Here are all permutations with their signatures
σ Sign.
[ 1 2 3 ] +
[ 1 3 2 ] − 1
[ 2 3 1 ] +
[ 2 1 3 ] − 1
[ 3 1 2 ] +
[ 3 2 1 ] − 1
ä We will denote by sig(σ) the signature of σ
Definition of Det.
det (A) =
σ
sig(σ)a 1 σ(1)a 2 σ(2) · · · anσ(n)
Where the sum runs over all (n!) possible permutations
of [1, 2 , , · · · , n].
Case n = 3 σ Sign. Det =
[ 1 2 3 ] +1 +a 11 a 22 a 33
[ 1 3 2 ] − 1 −a 11 a 23 a 32
[ 2 3 1 ] +1 +a 12 a 23 a 31
[ 2 1 3 ] − 1 −a 12 a 21 a 33
[ 3 1 2 ] +1 +a 13 a 21 a 32
[ 3 2 1 ] − 1 −a 13 a 22 a 31
