Study Guide
Block 3: S e l e c t e d Topics i n L i n e a r Algebra
Unit 5: Determinants
Overview
The u s u a l "pre-new-math"
t r a d i t i o n a l t r e a t m e n t of d e t e r m i n a n t s
i s q u i t e s t e r i l e and o f t e n s u b j e c t t o m i s i n t e r p r e t a t i o n . I t
seems t h a t t h e r e a s o n s f o r t e a c h i n g d e t e r m i n a n t s on t h e p r e c a l c u l u s l e v e l a r e o f t e n b e t t e r handled by t h e t e c h n i q u e of rowr e d u c i n g m a t r i c e s . Our aim i n t h i s u n i t i s t o e x p l a i n why
d e t e r m i n a n t s a r e important; what t h e y
a r e designed t o do; and
how one works w i t h them e f f i c i e n t l y .
I n a s e n s e , t h i s t r e a t m e n t could have come a t a d i f f e r e n t p l a c e
i n o u r c o u r s e , b u t w e p r e f e r t o i n t e r r u p t t h e theme developed
i n t h e p r e v i o u s u n i t i n o r d e r t o i n t r o d u c e m a t r i c e s now.
After
t h i s u n i t , w e s h a l l c o n t i n u e w i t h o u r s t u d y of l i n e a r t r a n s f o r m a t i o n s , b u t h o p e f u l l y w e s h a l l b e a b l e t o handle t h i n g s
much more e l e g a n t l y because o f o u r knowledge of d e t e r m i n a n t s .
Study Guide Block 3: Selected Topics in Linear Algebra Unit 5: Determinants 2.
Lecture 3.050 Study Guide
Block 3: S e l e c t e d Topics i n L i n e a r Algebra
Unit 5: Determinants
3 . Read:
Thomas, Appendix I . ( o p t i o n a l )
.
T h i s may be r e a d b e f o r e
o r a f t e r d o i n g t h e e x e r c i s e s , o r it may b e o m i t t e d e n t i r e l y i f
Our t h o u g h t i s t h a t t h e t r e a t m e n t i n t h e t e x t i s
you s o d e s i r e .
r e l a t i v e l y compact and complete ( a t l e a s t i n s o f a r a s t h e
v a r i o u s r e c i p e s a r e concerned)
.
Consequently, it may s e r v e t o
t i e t o g e t h e r any l o o s e ends t h a t might s t i l l seem t o be d a n g l i n g
t o you a f t e r o u r own t r e a t m e n t of t h e s u b j e c t .
4. Exercises :
3.5.1tL)
a. Compute
by w r i t i n g i t a s t h e sum o f two " s i m p l e r " 2 by 2 determinants.
b. U s e a s i m i l a r t e c h n i q u e t o e v a l u a t e
a s t h e sum of f o u r 2 by 2 d e t e r m i n a n t s .
C.
L e t A =[:+]and a. L e t
and
B
=[:;I.
I A ~ + ~ B ~ # \+A B I .
Show t h a t Study Guide
Block 3: S e l e c t e d T o p i c s i n L i n e a r Algebra
Unit 5: Determinants
3.5.2 (L) c o n t i n u e d
Form t h e m a t r i x AB and compute i t s d e t e r m i n a n t by u s i n g t h e
method o f p a r t ( b ) i n E x e r c i s e 3.5.1.
t h a t JABJ=
I n p a r t i c u l a r , show
I A ~ IBI.
b.
Show t h a t i f t h e n by n m a t r i x i s i n v e r t i b l e , t h e n
c.
V e r i f y p a r t ( b ) by d i r e c t computation i n t h e s p e c i f i c c a s e t h a t
3.5.3 (L)
Show t h a t i f P i s any i n v e r t i b l e n by n m a t r i x and A i s any n by
n matrix, then
3.5.4
1 PAP-' 1
IA] .
=
(optional) Let and
I A + 11
IAI
+
11lf-f
all
+ a 22 = 0.
a.
Show t h a t
b.
The t r a c e o f a s q u a r e m a t r i x i s t h e sum o f i t s d i a g o n a l elements.
-1
I f B i s i n v e r t i b l e , it t u r n s o u t t h a t B AB and A have t h e same
trace.
=
V e r i f y t h i s i n t h e s p e c i a l c a s e where A i s a s above and
Study Guide
Block 3: S e l e c t e d Topics i n L i n e a r Algebra
U n i t 5: Determinants
3.5.5(L)
Consider
a.
E v a l u a t e t h i s d e t e r m i n a n t by u s i n g t h e method o f c o - f a c t o r s
u n t i l it i s reduced t o a sum o f 2 by 2 d e t e r m i n a n t s .
b.
E v a l u a t e t h e same d e t e r m i n a n t b u t now by u s i n g row-reduction
techniques t h a t r e p l a c e a l l b u t t h e f i r s t e n t r y of t h e f i r s t
column by 0.
3.5.6
Evaluate
by t h e t e c h n i q u e o f ~ x e r c i s e3.5.5,
3.5.7
a.
(optional)
Prove d i r e c t l y from o u r axioms t h a t
( c o n t i n u e d on n e x t page)
p a r t (b).
Study Guide
Block 3: S e l e c t e d Topics i n L i n e a r Algebra
Unit 5: Determinants
3.5.7
continued
b.
Derive a s i m i l a r r e s u l t f o r
c.
Count t h e i n v e r s i o n s t o determine t h e v a l u e of
3.5.8
(optional) Use t h e v a r i o u s theorems a t our d i s p o s a l t o show t h a t a
.
l
r
l
l
r
r
1
2
3
2
2
2
r1 2
'
'3
= ( r 3 - r 2 )( r 3 - rl) ( r 2 - rl)
.
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Resource: Calculus Revisited: Complex Variables, Differential Equations, and Linear Algebra
Prof. Herbert Gross
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