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4
3
2
1
Outside the high school program
/ Matrix
19 exercises (100% corrected)
a
Newton's binomial formula
(4 exercices)
Newton's binomial formula and sequences
(1 exercice)
Matrices and complex numbers
(1 exercice)
Diagonalizable matrix
(7 exercices)
Systems overview
(5 exercices)
1.
Newton
’s
binomial
formula
E.5130
Consider
the
t
w
o
matrices
:
I
=
1
0
0
0
1
0
0
0
1
;
A
=
0
2
5
0
0
−
2
0
0
0
1
Determine
for
an
y
natural
n
um
b
er,
the
expression
of
the
p
o
w
ers
I
n
and
A
n
.
2
Deduce
the
expression
of
the
matrix
I
+
A
5
.
E.5162
Consider
the
matrix
A
defined
b
y
:
A
=
1
2
3
0
1
−
3
0
0
1
The
aim
of
this
exercise
is
to
giv
e
an
expression
for
A
n
for
an
y
natural
n
um
b
er
n
.
1
Consider
the
matrix
B
defined
b
y
B
=
A
−
I
3
.
Giv
e
the
expression
of
B
2
and
B
3
.
2
a
Can
w
e
use
Newton’s
binomial
form
ula
to
dev
elop
the
expression
I
3
+
B
n
for
n
non-zero
natural
n
um-
b
er?
Justify
y
our
answ
er.
b
Demonstrate
that
for
an
y
natural
n
um
b
er
n
greater
than
or
equal
to
3
,
w
e
ha
v
e
:
A
n
=
n
0
·
I
3
+
n
1
·
B
+
n
2
·
B
2
E.5163
Consider
the
t
w
o
matrices
A
and
D
de-
fined
b
y
:
A
=
−
1
1
2
0
−
1
−
2
0
0
−
1
;
D
=
−
1
0
0
0
−
1
0
0
0
−
1
The
aim
of
this
exercise
is
to
giv
e
an
expression
for
A
n
for
an
y
natural
n
um
b
er
n
.
1
Consider
the
matrix
B
defined
b
y
B
=
A
−
D
.
Giv
e
the
expression
of
B
2
and
B
3
.
2
a
Can
w
e
use
Newton’s
binomial
form
ula
to
dev
elop
the
expression
D
+
B
n
for
n
non-zero
natural
n
um-
b
er?
Justify
y
our
answ
er.
b
Demonstrate
that
for
an
y
natural
n
um
b
er
n
greater
than
or
equal
to
3
,
w
e
ha
v
e
:
A
n
=
(
−
1)
n
·
n
0
·
I
3
+
(
−
1)
n
−
1
·
n
1
·
B
+
(
−
1)
n
−
2
·
n
2
·
B
2
E.5131
Consider
the
matrix
A
defined
b
y
:
A
=
3
1
2
0
3
−
1
0
0
3
1
W
rite
the
matrix
A
in
the
form
B
+
C
where
B
is
a
square
diagonal
matrix.
2
a
F
or
an
y
in
teger
n
,
giv
e
the
expression
for
C
n
.
b
V
erify
that
the
matrices
B
and
C
are
comm
utativ
e.
c
Establish
the
follo
wing
equalit
y
for
an
y
in
teger
n
greater
than
or
equal
to
2
:
A
n
=
B
n
+
n
·
C
·
B
n
+
n
2
·
C
2
·
B
n
−
2
2.
Newton
’s
binomial
formula
and
se
quenc
es
E.5164
Consider
the
A
square
matrix
of
dimen-
sion
3
defined
b
y
:
A
=
2
0
3
0
1
0
0
0
2
1
Consider
the
sequence
u
n
defined
b
y
:
u
0
=
0
;
u
n
+1
=
2
·
u
n
+
3
×
2
n
for
all
n
∈
N
Establish
that
for
an
y
in
teger
n
,
w
e
ha
v
e
:
A
n
=
2
n
0
u
n
0
1
0
0
0
2
n
2
Consider
the
t
w
o
matrices
D
and
S
defined
b
y
:
D
=
2
0
0
0
1
0
0
0
2
;
S
=
0
0
3
0
0
0
0
0
0
a
Sho
w
that
:
D
·
S
=
S
·
D
.
b
Sho
w
that
for
an
y
k
2
,
w
e
ha
v
e
:
S
k
=
0
3
.
c
Using
Newton’s
binomial
form
ula,
determine
the
ex-
pression
of
A
n
for
an
y
non-zero
natural
n
um
b
er.
3
Deduce
the
explicit
form
ula
for
the
sequence
u
n
.
3.
Matric
es
and
c
omplex
numb
ers
E.5447
Consider
the
square
matrix
A
of
order
2
defined
b
y
:
A
=
1
−
1
2
−
1
Consider
the
matrices
P
and
Q
defined
b
y
:
P
=
1
1
1
+
i
1
−
i
;
Q
=
1
2
+
1
2
·
i
−
1
2
·
i
1
2
−
1
2
·
i
1
2
·
i
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5130
sacados/5130
chapExoCorrec/5162
sacados/5162
chapExoCorrec/5163
sacados/5163
chapExoCorrec/5131
sacados/5131
chapExoCorrec/5164
sacados/5164
chapExoCorrec/5447
sacados/5447
where
i
is
the
complex
n
um
b
er
v
erifying
:
i
2
=
−
1
.
1
Sho
w
that
the
matrices
P
and
Q
are
in
v
erses
of
eac
h
other.
2
Sho
w
that
matrix
P
−
1
·
A
·
P
is
a
diagonal
matrix.
W
e
note
D
this
diagonal
matrix
in
the
rest
of
the
exercise.
3
a
Establish,
using
reasoning
b
y
recurrence,
the
follo
w-
ing
relationship
for
an
y
natural
n
um
b
er
n
:
A
n
=
P
·
D
n
·
P
−
1
b
Justify
that
for
an
y
in
teger
n
,
w
e
ha
v
e
the
relation
A
n
+4
=
A
n
4.
Diagonalizable
matrix
E.5161
Consider
the
matrices
A
and
P
defined
b
y
:
A
=
5
3
−
6
−
4
;
P
=
1
−
1
−
1
2
1
Sho
w
that
the
matrix
P
is
an
in
v
ertible
matrix
and
giv
e
the
expression
for
the
matrix
P
−
1
.
2
a
Establish
the
existence
of
a
matrix
D
v
erifying
the
equalit
y
:
A
=
P
·
D
·
P
−
1
b
Using
reasoning
b
y
recurrence,
establish
that
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
the
equalit
y
:
A
n
=
P
·
D
n
·
P
−
1
c
Deduce,
for
an
y
non-zero
natural
n
um
b
er
n
,
the
ex-
pression
for
A
n
.
E.5160
Consider
the
matrices
A
and
P
defined
b
y
:
A
=
4
3
−
2
−
1
;
P
=
3
1
−
2
−
1
1
Sho
w
that
the
matrix
P
is
an
in
v
ertible
matrix
and
giv
e
the
expression
of
the
matrix
P
−
1
.
2
a
Establish
the
follo
wing
equalit
y
:
A
=
P
·
2
0
0
1
·
P
−
1
b
Deduce,
for
an
y
non-zero
natural
n
um
b
er
n
,
the
ex-
pression
for
A
n
.
E.5158
Consider
the
matrix
A
defined
b
y
:
A
=
2
2
−
1
−
1
−
1
1
−
2
−
4
3
and
the
t
w
o
matrices
P
and
Q
defined
b
y
:
P
=
1
−
3
−
2
−
1
1
1
−
2
−
1
0
;
Q
=
1
2
−
1
−
2
−
4
1
3
7
−
2
1
Sho
w
that
the
matrices
P
and
Q
are
t
w
o
matrices
that
are
in
v
erses
of
eac
h
other.
2
W
e
denote
D
the
matrix
defined
b
y
:
I
=
2
0
0
0
1
0
0
0
1
a
Establish
equalit
y
:
A
=
P
·
D
·
Q
b
Establish,
using
reasoning
b
y
recurrence,
that
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
A
n
=
P
·
D
n
·
Q
c
Deduce,
for
an
y
non-zero
natural
n
um
b
er
n
,
the
ex-
pression
of
the
matrix
A
n
.
E.5159
Consider
the
matrix
A
defined
b
y
:
A
=
−
2
−
1
−
3
−
7
4
−
9
1
1
2
and
the
t
w
o
matrices
P
and
Q
defined
b
y
:
P
=
1
−
2
−
1
−
1
−
1
2
−
1
1
1
;
Q
=
−
3
1
−
5
−
1
0
−
1
−
2
1
−
3
1
Sho
w
that
the
matrices
P
and
Q
are
t
w
o
matrices
that
are
in
v
erses
of
eac
h
other.
2
Establish
the
existence
of
a
matrix
D
v
erifying
the
equal-
it
y
:
A
=
P
·
D
·
Q
and
admitting
as
expression
:
D
=
a
0
0
0
b
0
0
0
c
where
a
,
b
,
c
are
three
real
n
um-
b
ers.
3
a
Using
reasoning
b
y
recurrence,
establish
that
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
the
equalit
y
:
A
n
=
P
·
D
n
·
Q
b
Deduce,
for
an
y
non-zero
natural
n
um
b
er
n
,
the
ex-
pression
for
the
matrix
A
n
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5161
sacados/5161
chapExoCorrec/5160
sacados/5160
chapExoCorrec/5158
sacados/5158
chapExoCorrec/5159
sacados/5159
E.5146
Consider
the
matrix
A
square
of
dimen-
sion
3
defined
b
y
:
A
=
3
−
6
−
1
4
−
5
−
2
2
−
6
0
The
aim
of
the
exercise
is
to
determine
an
expression
for
the
matrix
A
n
for
an
y
natural
n
um
b
er
n
.
T
o
do
this,
consider
the
t
w
o
square
matrices
P
and
Q
:
P
=
1
−
3
−
2
0
−
2
−
1
2
−
3
−
2
;
Q
=
−
1
0
1
2
−
2
−
1
−
4
3
2
1
Sho
w
that
the
matrices
P
and
Q
are
t
w
o
matrices
that
are
in
v
erses
of
eac
h
other.
2
Establish
equalit
y
:
A
=
P
·
1
0
0
0
−
2
0
0
0
−
1
·
Q
Note
D
the
matrix
:
D
=
1
0
0
0
−
2
0
0
0
−
1
3
a
Giv
e
the
expression
for
the
matrix
D
n
where
n
is
a
natural
n
um
b
er.
b
Establish
for
an
y
natural
n
um
b
er
n
,
the
equalit
y
:
A
n
=
P
·
D
n
·
Q
c
Giv
e
the
expression
of
A
n
as
a
function
of
n
.
E.5356
Consider
the
t
w
o
matrices
of
dimensions
2
:
A
=
−
1
−
6
1
4
;
P
=
3
2
−
1
−
1
1
a
Justify
that
the
matrix
P
is
an
in
v
ertible
matrix.
b
Giv
e
the
expression
for
the
matrix
P
−
1
in
v
erse
of
the
matrix
P
.
2
Determine
the
v
alue
of
the
reals
¸
and
˛
ac
hieving
the
follo
wing
equalit
y
:
A
=
P
·
¸
0
0
˛
·
P
−
1
3
a
Using
reasoning
b
y
recurrence,
establish
the
follo
w-
ing
equalit
y
for
an
y
natural
n
um
b
er
n
:
A
n
=
P
·
¸
n
0
0
˛
n
·
P
−
1
b
Giv
e
the
expression
of
A
n
as
a
function
of
n
.
E.6099
Consider
the
t
w
o
matrices
:
A
=
4
1
−
2
1
;
P
=
1
1
−
1
−
2
1
a
Justify
that
the
matrix
P
is
in
v
ertible
and
giv
e
the
expression
for
the
in
v
erse
matrix
of
P
.
b
Let’s
note
D
=
P
−
1
·
A
·
P
.
Sho
w
that
the
matrix
D
is
diagonal.
2
a
Using
reasoning
b
y
recurrence,
sho
w
that
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
A
n
=
P
·
D
n
·
P
−
1
b
Giv
e
the
expression
of
the
matrix
A
n
as
a
function
of
the
non-zero
natural
n
um
b
er
n
.
5.
Systems
overview
E.6118
A
t
a
coffee
shop,
here
are
t
w
o
orders
and
their
bill
amoun
ts
:
In
v
oice
:
4.5
e
.
In
v
oice
:
10.8
e
In
this
café,
what
are
the
prices
of
a
croissan
t
and
a
coffee?
E.6119
A
t
a
coffee
shop,
here
are
t
w
o
orders
and
their
bill
amoun
ts
:
In
v
oice
:
11
e
.
In
v
oice
:
14.3
e
In
this
café,
what
are
the
prices
of
a
croissan
t
and
a
can?
E.6120
A
t
a
cafe,
here
are
t
w
o
orders
and
their
bill
amoun
ts
:
In
v
oice
:
15.2
e
.
In
v
oice
:
10
e
What
are
the
prices,
in
this
café,
for
a
coffee
and
a
can?
E.6121
1
Solv
e
the
follo
wing
system
using
the
linear
com
bination
metho
d
:
(
S
)
:
3
x
+
2
y
=
5
4
x
+
3
y
=
3
2
Solv
e
the
follo
wing
system
using
the
substitution
metho
d
:
(
T
)
:
3
x
+
y
=
16
8
x
−
5
y
=
12
E.6122
Consider
the
system
(
S
)
of
equations
:
x
−
3
y
=
8
4
x
+
y
=
−
7
Solv
e
the
system
(
S
)
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5146
sacados/5146
chapExoCorrec/5356
sacados/5356
chapExoCorrec/6099
sacados/6099
chapExoCorrec/6118
sacados/6118
chapExoCorrec/6119
sacados/6119
chapExoCorrec/6120
sacados/6120
chapExoCorrec/6121
sacados/6121
chapExoCorrec/6122
sacados/6122
6.
Unclassifie
d
financial
ye
ars
E.5392
W
e
consider
an
ev
olutionary
pro
cess
ad-
mitting
the
follo
wing
transition
matrix
:
T
=
7
10
−
3
10
1
5
1
5
1
W
e
consider
the
matrix
P
définie
b
y
:
P
=
3
1
2
1
a
Sho
w
that
the
matrix
P
est
is
an
in
v
ertible
matrix
and
determine
the
expression
of
its
in
v
erse
matrix.
b
Determine
the
existence
of
a
D
diagonal
matrix
v
eri-
fying
the
equalit
y
:
T
=
P
·
D
·
P
−
1
.
c
Établir,
à
l’aide
d’un
raisonnemen
t
par
récurrence,
la
relation
suiv
an
te
p
our
tout
en
tier
naturel
n
:
T
n
=
P
·
D
n
·
P
−
1
.
d
Determine
the
expression
of
the
matrix
T
n
as
a
func-
tion
of
the
natural
n
um
b
er
n
.
2
Consider
the
t
w
o
sequences
a
n
and
b
n
with
resp
ec-
tiv
e
first
terms
a
0
and
b
0
and
satisfying
the
relation
:
a
n
+1
b
n
+1
=
T
·
a
n
b
n
a
Using
recursiv
e
reasoning,
establish
the
follo
wing
equalit
y
for
all
natural
n
um
b
ers
n
:
a
n
b
n
=
T
n
·
a
0
b
0
b
Determine
an
expression
for
the
terms
of
these
t
w
o
sequences
in
terms
of
n
,
a
0
and
b
0
.
c
Deduce
the
limit
of
these
t
w
o
sequences.
h
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