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National and Regional Contests
India Contests
India LIMIT
2019 LIMIT
2019 LIMIT Category C
Problem 9
idempotent matrices
idempotent matrices
Source: LIMIT 2019 CCS2 P9
April 28, 2021
Matrices
linear algebra
matrix
Problem Statement
P
∈
A
n
(
R
)
=
{
M
n
×
n
∣
M
2
=
M
}
P\in A_n(\mathbb R)=\{M_{n\times n}|M^2=M\}
P
∈
A
n
(
R
)
=
{
M
n
×
n
∣
M
2
=
M
}
. Which of the following are true?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
P
T
=
P
,
∀
P
∈
A
n
(
R
)
<span class='latex-bold'>(A)</span>~P^T=P,\forall P\in A_n(\mathbb R)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
P
T
=
P
,
∀
P
∈
A
n
(
R
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
∃
P
≠
0
,
P
∈
A
n
(
R
)
with
tr
(
P
)
=
0
<span class='latex-bold'>(B)</span>~\exists P\ne0,P\in A_n(\mathbb R)\text{ with }\operatorname{tr}(P)=0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
∃
P
=
0
,
P
∈
A
n
(
R
)
with
tr
(
P
)
=
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
∃
X
n
×
r
such that
P
x
=
X
for
r
=
rank
(
P
)
<span class='latex-bold'>(C)</span>~\exists X_{n\times r}\text{ such that }Px=X\text{ for }r=\operatorname{rank}(P)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
∃
X
n
×
r
such that
P
x
=
X
for
r
=
rank
(
P
)
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