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MA 36600 FINAL REVIEW
3
i.
A
is invertible i.e., there exists
B
such that
AB
=
BA
=
I
.
ii.
A
is nonsingular i.e., the
x
=
0
is the only solution to
Ax
=
0
.
iii. det
A
°
= 0.
•
Say that we are given an
m
×
n
matrix where the entries are functions of time
t
:
A
(
t
)=
°
a
ij
(
t
)
±
We deFne the
derivative
and
(indefnite) integral
as those
m
×
n
matrices
d
dt
A
(
t
)=
°
d
dt
a
ij
(
t
)
±
and
²
t
A
(
τ
)
dτ
=
°²
t
a
ij
(
τ
)
dτ
±
.
We have the Product Rule i.e., given an
m
×
p
matrix
A
=
A
(
t
) and a
p
×
n
matrix
B
=
B
(
t
),
d
dt
³
A
(
t
)
B
(
t
)
´
=
A
(
t
)
d
B
dt
(
t
)+
d
A
dt
(
t
)
B
(
t
)
.
We have the
Fundamental Theorem o± Calculus
i.e., given an
m
×
n
matrix
A
=
A
(
t
),
d
dt
²
t
A
(
τ
)
dτ
=
A
(
t
)
.
§
7.3: Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors.
•
A system of Frst order linear di±erential equations can be expressed in the form
d
dt
x
1
x
2
.
.
.
x
m
µ
¶·
¸
m
dim’l vector
=
p
11
(
t
)
p
12
(
t
)
···
p
1
n
(
t
)
p
21
(
t
)
p
22
(
t
)
···
p
2
n
(
t
)
.
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue UniversityWest Lafayette.
 Spring '09
 EdrayGoins

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