Topic 3 (3) Determinants.pdf determinants properties

Properties of determinants

1. Determinants
Properties of determinants and Cramer’s
rule

2. ThislectureiscuratedbasedonNumerical
AnalysisbyAnton,HandKaul,A(2019).
ElementaryLinearAlgebra,12thed.Pleaserefer
tothetextbook(officialtextbook)forfull
details.
Thetopicsdiscussedinthislecturecanbefound
inSection2.3ofthetextbook.

3. In this section, we use determinants to:
•derive a formula for the inverse of an invertible matrix
•derive formulas for the solutions of certain kinds of linear
systems.

4. Let ?be an JHJmatrix.
Property 1:
@APG?LG
?
?.
Basic properties of determinants
Quick illustration:

5. Let ?be an JHJmatrix.
Property 1:
@APG?LG
?
?.
Property 2 (non-property):
@AP?E?M@AP?E@AP?.
Basic properties of determinants
Quick illustration:

6. Let ?be an JHJmatrix.
Property 1:
@APG?LG
?
?.
Property 2 (non-property):
@AP?E?M@AP?E@AP?.
Property 3:
@AP??L@AP?@AP?.
Basic properties of determinants
Quick illustration:

7. Let ?be an JHJmatrix.
Property 1:
@APG?LG
?
?.
Property 2 (non-property):
@AP?E?M@AP?E@AP?.
Property 3:
@AP??L@AP?@AP?.
Property 4:
If ?is invertible, @AP?
?5
Ls@AP?? .
Basic properties of determinants
Quick illustration/proof:
Since ?
?5
?L?
?
, it follows that
@AP?
?5
?L@AP?
?
Ls?
Also, from Property 3:
@AP?
?5
?L@AP?
?5
@AP?Ls?
Since @AP?Mr:@AP?
?5
Ls@AP?? .

8. Entries and Cofactors from Different Rows
Let
?L
utFs
sxu
tFvr
?
The cofactors of ?are:
%
55
L
%
56
L
%
57
L
%
65
L
%
66
L
%
67
L
%
75
L
%
76
L
%
77
L

9. Observation 1:
Cofactor expansion of @AP? along the first row is
@AP?L=
55
%
55
E=
56
%
56
E=
57
%
57
L u s tE t xEF s F s xL x v?
Cofactor expansion of @AP? along the first column is
@AP?L=
55
%
55
E=
65
%
65
E=
75
%
75
L u s tE s vE t s tL x v?
Entries and Cofactors from Different Rows
Let
?L
u tF s
s x u
tF v r
?
The cofactors of ?are:
%
55
L s t
%
56
L x
%
57
LF s x
%
65
L v
%
66
L t
%
67
L s x
%
75
L s t
%
76
LF s r
%
77
L s x

10. Observation 1:
Cofactor expansion of @AP? along the first row is
@AP?L=
55
%
55
E=
56
%
56
E=
57
%
57
L u s tE t xEF s F s xL x v?
Cofactor expansion of @AP? along the first column is
@AP?L=
55
%
55
E=
65
%
65
E=
75
%
75
L u s tE s vE t s tL x v?
Entries and Cofactors from Different Rows
Let
?L
u tF s
s x u
tF v r
?
The cofactors of ?are:
%
55
L s t
%
56
L x
%
57
LF s x
%
65
L v
%
66
L t
%
67
L s x
%
75
L s t
%
76
LF s r
%
77
L s x
Cofactor expansion along any rows or columns give the same
value which is @AP?

11. Observation 2:
Multiply the entries in the first row by the corresponding
cofactors from the second row and add the resulting
products
=
55
%
65
E=
56
%
66
E=
57
%
67
Lu vEt tEFs sxLr?
Multiply the entries in the first column by the
corresponding cofactors from the second column and
add the resulting products
=
55
%
56
E=
65
%
66
E=
75
%
76
Lu xEs tEt FsrLr?
Entries and Cofactors from Different Rows
Let
?L
utFs
sxu
tFvr
?
The cofactors of ?are:
%
55
Lst
%
56
Lx
%
57
LFsx
%
65
Lv
%
66
Lt
%
67
Lsx
%
75
Lst
%
76
LFsr
%
77
Lsx

12. Observation 2:
Multiply the entries in the first row by the corresponding
cofactors from the second row and add the resulting
products
=
55
%
65
E=
56
%
66
E=
57
%
67
Lu vEt tEFs sxLr?
Multiply the entries in the first column by the
corresponding cofactors from the second column and
add the resulting products
=
55
%
56
E=
65
%
66
E=
75
%
76
Lu xEs tEt FsrLr?
Entries and Cofactors from Different Rows
Let
?L
utFs
sxu
tFvr
?
The cofactors of ?are:
%
55
Lst
%
56
Lx
%
57
LFsx
%
65
Lv
%
66
Lt
%
67
Lsx
%
75
Lst
%
76
LFsr
%
77
Lsx
If the =’s and %
??
’s come from different rows/columns, the sum is zero.

13. Definition:
If ?L?
??
is an JHJmatrix, and %
??
is the cofactor of
=
??
, then the matrix
%
55
%
56
%
65
%
66
?%
5?
?%
6?
??
%
?5
%
?6
??
?%
??
?
is called the matrix of cofactors from ?.
The transpose of this matrix is called the adjoint of ?and
is denoted by =@F?.
Adjoint of a matrix

14. Definition:
If ?L?
??
is an JHJmatrix, and %
??
is the cofactor of
=
??
, then the matrix
%
55
%
56
%
65
%
66
?%
5?
?%
6?
??
%
?5
%
?6
??
?%
??
?
is called the matrix of cofactors from ?.
The transpose of this matrix is called the adjoint of ?and
is denoted by =@F?.
Adjoint of a matrix
Example:
Let
?L
utFs
sxu
tFvr
?
The cofactors of ?are:
%
55
Lst?%
56
Lx?%
57
LFsx?
%
65
Lv?%
66
Lt?%
67
Lsx?
%
75
Lst?%
76
LFsr?%
77
Lsx?
Matrix of cofactorsL ,
=@F?L .

15. Theorem:
If ?is invertible, then
?
?5
L
s
@AP?
=@F??
Inverse of matrix using adjoint

16. Theorem:
If ?is invertible, then
?
?5
L
s
@AP?
=@F??
Inverse of matrix using adjoint
Proof:
Consider

17. Theorem:
If ?is invertible, then
?
?5
L
s
@AP?
=@F??
Inverse of matrix using adjoint
Proof:
Consider
?=@F?
??
L=
?5
%
?5
E=
?6
%
?6
E?E=
??
%
??
The entry of ?=@F?in the Ethrow and Fthcolumn:

18. Theorem:
If ?is invertible, then
?
?5
L
s
@AP?
=@F??
Inverse of matrix using adjoint
Proof:
Consider
?=@F?
??
L=
?5
%
?5
E=
?6
%
?6
E?E=
??
%
??
The entry of ?=@F?in the Ethrow and Fthcolumn:
RECALL:
Observation 1 --Cofactor expansion along any
rows or columns give the same value which is
@AP?
Observation 2 --If the =’s and %
??
’s come from
different rows/columns, the sum is zero.

19. Theorem:
If ?is invertible, then
?
?5
L
s
@AP?
=@F??
Inverse of matrix using adjoint
Proof:
Consider
?=@F?
??
L=
?5
%
?5
E=
?6
%
?6
E?E=
??
%
??
The entry of ?=@F?in the Ethrow and Fthcolumn:
RECALL:
Observation 1 --Cofactor expansion along any
rows or columns give the same value which is
@AP?
Observation 2 --If the =’s and %
??
’s come from
different rows/columns, the sum is zero.
If ELF:
If EMF:
?=@F?
??
L=
?5
%
?5
E=
?6
%
?6
E?E=
??
%
??
L@AP?
?=@F?
??
L=
?5
%
?5
E=
?6
%
?6
E?E=
??
%
??
Lr.

20. Theorem:
If ?is invertible, then
?
?5
L
s
@AP?
=@F??
Inverse of matrix using adjoint
Proof:
Consider
L
@AP? r
r@AP?
? r
??
? ?
r r
? r
?@AP?
L@AP??
?

21. Theorem:
If ?is invertible, then
?
?5
L
s
@AP?
=@F??
Inverse of matrix using adjoint
Proof:
Consider
L
@AP? r
r@AP?
? r
??
? ?
r r
? r
?@AP?
L@AP??
?
It follows that
?=@F?L@AP??
?
?
Multiply through by
5
????
?
?5
to give
?
?5
L
s
@AP?
=@F??

22. Theorem:
If ?is invertible, then
?
?5
L
s
@AP?
=@F??
Inverse of matrix using adjoint
Example:
Let
?L
utFs
sxu
tFvr
?
=@F?L
stvst
xtFsr
Fsxsxsx
.
?
?5
L
5
????
=@F?L
5
:8
stvst
xtFsr
Fsxsxsx
L
-.
20
0
20
-.
20
2
20
.
20
F
-,
20
F
-2
20
-2
20
-2
20
.

23. Theorem:
If ?is invertible, then
?
?5
L
s
@AP?
=@F??
Inverse of matrix using adjoint
Example:
Let
?L
utFs
sxu
tFvr
?
=@F?L
stvst
xtFsr
Fsxsxsx
.
?
?5
L
5
????
=@F?L
5
:8
stvst
xtFsr
Fsxsxsx
L
-.
20
0
20
-.
20
2
20
.
20
F
-,
20
F
-2
20
-2
20
-2
20
.
A WORD OF CAUTION:
Althoughitisnicetohaveaformulafor
theinverse,computingitaccuratelyvia
theformulacanbeachallenge,
especiallywhen@AP?Nr.

24. Theorem:
If ? Lis a system of Jlinear equations in J
unknowns such that @AP?Mr, then the system
has a unique solution. This solution is
T
5
L
@AP?
5
@AP?
?T
6
L
@AP?
6
@AP?
???T
?
L
@AP?
?
@AP?
?
where ?
?
is the matrix obtained by replacing the
entries in the Fthcolumn of ?by the entries of .
Cramer’s rule

25. Theorem:
If ? Lis a system of Jlinear equations in J
unknowns such that @AP?Mr, then the system
has a unique solution. This solution is
T
5
L
@AP?
5
@AP?
?T
6
L
@AP?
6
@AP?
???T
?
L
@AP?
?
@AP?
?
where ?
?
is the matrix obtained by replacing the
entries in the Fthcolumn of ?by the entries of .
Cramer’s rule
Example:

26. PleasegothroughExercisesSet2.3inthetext
booktogiveyouthepracticeyouneedto
enhanceyourunderstanding.
Youwillfindbriefsolutionstooddnumbered
questionsattheendofthetextbook.ONLY
REFERTOTHESOLUTIONSAFTERYOUHAVE
MADEATTEMPTSONTHEEXERCISES.
Tutorialquestionswillbepreparedbasedon
questionfromthoseexercises,butwillbe
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