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IMC
1995 IMC
1
1
Part of
1995 IMC
Problems
(1)
IMC 1995 Problem 1
Source: IMC 1995
2/18/2021
Let
X
X
X
be a invertible matrix with columns
X
1
,
X
2
.
.
.
,
X
n
X_{1},X_{2}...,X_{n}
X
1
,
X
2
...
,
X
n
. Let
Y
Y
Y
be a matrix with columns
X
2
,
X
3
,
.
.
.
,
X
n
,
0
X_{2},X_{3},...,X_{n},0
X
2
,
X
3
,
...
,
X
n
,
0
. Show that the matrices
A
=
Y
X
−
1
A=YX^{-1}
A
=
Y
X
−
1
and
B
=
X
−
1
Y
B=X^{-1}Y
B
=
X
−
1
Y
have rank
n
−
1
n-1
n
−
1
and have only
0
0
0
´s for eigenvalues.
linear algebra
matrix