Study Guide
B l o c k 4: Matrix A l g e b r a
U n i t 2:
1.
I n t r o d u c t i o n t o Matrix A l g e b r a
L e c t u r e 4.020
Study Guide
Block 4: M a t r i x Algebra
U n i t 2 : I n t r o d u c t i o n t o M a t r i x Alqebra
Lecture 4.020
continued
Study Guide
Block 4 : M a t r i x Algebra
U n i t 2 : I n t r o d u c t i o n t o M a t r i x Algebra
2.
Read: Supplementary N o t e s , Chapter 6', S e c t i o n D
3.
( O p t i o n a l ) Read: Thomas, S e c t i o n 13.2
Exercises
Show t h a t A ( B
+
C) = AB
+
AC and A ( B C ) = ( A B ) C where A , B ,
and
C a r e each 2 x 2 matrices.
)
1 1
Let A =
i0.
0
and 0 =
0
o)
a.
F i n d a l l m a t r i c e s B s u c h t h a t AB = 0 .
b.
Find a l l m a t r i c e s C s u c h t h a t AC = 0 b u t CA f 0.
Let A =
(:
-:)
a nd 0
=(: i).
Find a l l m a t r i c e s B s u c h t h a t AB = 0
Let A =
(: -:), =(: :)
B
and I
=('
0
a.
F i n d a m a t r i x X s u c h t h a t AX = I
b.
Does t h e r e e x i s t a m a t r i x X s u c h t h a t BX = IZ
For any s q u a r e m a t r i x A, we d e f i n e
r i s any p o s i t i v e i n t e g e r ,
a.
Compute
if A =
b.
Compute
if A =
c3
-
t o mean AA.. .A,
where
n times
i: 1)
( c o n t i n u e d on t h e n e x t page)
4.2.3
Study Guide
Block 4: M a t r i x Algebra
U n i t 2: I n t r o d u c t i o n t o M a t r i x Algebra
-
4.2.5
(continued)
.
Compute A
Let A
and '
A
=(: )
if A
and l e t I =
1
(i
--
:)
Compute
-
b.
Compute
u s i n g t h e r e s u l t of p a r t ( a ) .
c.
Compute A
-
-
3
a.
2A
--
=(-: -:-:)
1
c.
-
24 I
7
4.2.7
Let A =
( )
.
Find a l l m a t r i c e s I3 s u c h t h a t AB = BA.
4.2.8
I f A i s any m a t r i x , w e d e f i n e t h e t r a n s p o s e of A, w r i t t e n A
T
t o b e t h e matrix o b t a i n e d when we i n t e r c h a n g e t h e rows and
columns of A.
T h a t is, t h e columns of A a r e t h e rows of A
2
What is
if A =
T T
I f A i s any m a t r i x , compute (A )
.
I f A and B a r e 2 x 2 m a t r i c e s show t h a t
T
Find a l l 2 x 2 matrices A f o r which AAT = A A.
T
.
,
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Resource: Calculus Revisited: Multivariable Calculus
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