MATH 0047
N
Determinants
Definition a permutation of a set of numbers I n is an
arrangement of the numbers in some order without
commissions or repetitions
,the possible permutations of set 1,213 is
112,3 1 3,2 2 1,3 2,311 3 1,2 3,211
there are n permutations of the set where n is the
number values
of different
Inversion Occurs in a permutation when a
larger number
precedes a smaller one
If O 1 2,3 there are 0 inversions
If o 2,1 3 there is 1 inversion
O 3,2 1 there are 3 inversions
If
Sign The sign of a permutation is defined as
sign o 1 if there is an even number
inversions of
sign o 1 if there is an odd number of
inversions
An elementary product of matrix
Elementary product is a product of n entries
an nxn
such that
there is one each
row and exactly
column entry from
signed elementary product
sign o x elementary product
Determinant A Square Matrix
of
diet A Ee sign o Aloe Anon
this is the sum
of all signed elementary products
,Co factor Expansion
obtain the minor of a certain Ali by removing
column i and j and computing
entry
the determinant the
entries of
remaining
i e
A 11 11
minor of ALI I A22A33 A23A32
co
factor C1 tix A II
Determinant Z cofactor x
entry value
it
El 1 Ai Ali in the raw column that
we are observing
Note
there is zero column in the matrix the determina
If
will be
a
zero
row
always
det A det AT
upper triangle A square matrix is upper triangular it the
entries above the main diagonal art zeros i e
1
Lower A matrix is lower the
triangle square
entries below the
triangular
main diagonal are if zeros i e
I
The determinants of an upper triangular and lower triangular
matrix is the product the entries on the main diagonal
of
A matrix is invertible if the determinent is a non zero
, ma it s er b r e e er e n n ze
f
Determinants
of Elementary Matrices
det Dn i x A A det A
det En i d A det A
det Pn i d A det A
equivalently
det A
I det Dn fi d A
det A det En i a A
det A det Pn i A
of EA
overall that the determinant is to the
product of
we
find
the determinant the
equal
matrix and
the determinant A of elementary
of
det EA dete x detLA
Theorum 2.23 A square
A
det
min
O
matrix A is invertible if and only
if
Theorum 2.24 For any square nxn matrices A and B
det AB det A x det B
Vector Spaces
suppose that a b are vectors in a vector space then
the sum a and b is defined
by
of
at b i ait bi
an
1 11.1 1 1
N
Determinants
Definition a permutation of a set of numbers I n is an
arrangement of the numbers in some order without
commissions or repetitions
,the possible permutations of set 1,213 is
112,3 1 3,2 2 1,3 2,311 3 1,2 3,211
there are n permutations of the set where n is the
number values
of different
Inversion Occurs in a permutation when a
larger number
precedes a smaller one
If O 1 2,3 there are 0 inversions
If o 2,1 3 there is 1 inversion
O 3,2 1 there are 3 inversions
If
Sign The sign of a permutation is defined as
sign o 1 if there is an even number
inversions of
sign o 1 if there is an odd number of
inversions
An elementary product of matrix
Elementary product is a product of n entries
an nxn
such that
there is one each
row and exactly
column entry from
signed elementary product
sign o x elementary product
Determinant A Square Matrix
of
diet A Ee sign o Aloe Anon
this is the sum
of all signed elementary products
,Co factor Expansion
obtain the minor of a certain Ali by removing
column i and j and computing
entry
the determinant the
entries of
remaining
i e
A 11 11
minor of ALI I A22A33 A23A32
co
factor C1 tix A II
Determinant Z cofactor x
entry value
it
El 1 Ai Ali in the raw column that
we are observing
Note
there is zero column in the matrix the determina
If
will be
a
zero
row
always
det A det AT
upper triangle A square matrix is upper triangular it the
entries above the main diagonal art zeros i e
1
Lower A matrix is lower the
triangle square
entries below the
triangular
main diagonal are if zeros i e
I
The determinants of an upper triangular and lower triangular
matrix is the product the entries on the main diagonal
of
A matrix is invertible if the determinent is a non zero
, ma it s er b r e e er e n n ze
f
Determinants
of Elementary Matrices
det Dn i x A A det A
det En i d A det A
det Pn i d A det A
equivalently
det A
I det Dn fi d A
det A det En i a A
det A det Pn i A
of EA
overall that the determinant is to the
product of
we
find
the determinant the
equal
matrix and
the determinant A of elementary
of
det EA dete x detLA
Theorum 2.23 A square
A
det
min
O
matrix A is invertible if and only
if
Theorum 2.24 For any square nxn matrices A and B
det AB det A x det B
Vector Spaces
suppose that a b are vectors in a vector space then
the sum a and b is defined
by
of
at b i ait bi
an
1 11.1 1 1
