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SEEMOUS
2020 SEEMOUS
Problem 3
Problem 3
Part of
2020 SEEMOUS
Problems
(1)
Rank of Matrix and Its Trance
Source: SEEMOUS 2020 P3
5/2/2020
Let
n
n
n
be a positive integer,
k
∈
C
k\in \mathbb{C}
k
∈
C
and
A
∈
M
n
(
C
)
A\in \mathcal{M}_n(\mathbb{C})
A
∈
M
n
(
C
)
such that
Tr
A
≠
0
\text{Tr } A\neq 0
Tr
A
=
0
and
rank
A
+
rank
(
(
Tr
A
)
⋅
I
n
−
k
A
)
=
n
.
\text{rank } A +\text{rank } ((\text{Tr } A) \cdot I_n - kA) =n.
rank
A
+
rank
((
Tr
A
)
⋅
I
n
−
k
A
)
=
n
.
Find
rank
A
\text{rank } A
rank
A
.
linear algebra
matrix