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MATH0047 - Advanced Linear Algebra
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lecture revision notes for the 2nd year course module MATH0047 - Advanced Linear Algebra. Shortened and simplified to use as revision summary for exams

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  • July 21, 2022
  • 17
  • 2021/2022
  • Lecture notes
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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

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