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Mathematics
Problem
Let
A
A
A
be an
n
×
n
n\times n
n
×
n
matrix such that
A
2
=
2
A
A^{2}=2A
A
2
=
2
A
. (i) Show that
A
(
I
n
−
A
)
=
I
n
A
−
A
2
A(I_n-A)=I_nA-A^2
A
(
I
n
−
A
)
=
I
n
A
−
A
2
. (ii) Using the result of part (i), prove that the matrix
I
n
−
A
I_n-A
I
n
−
A
is invertible. (iii) Determine
(
I
n
−
A
)
−
1
(I_n-A)^{-1}
(
I
n
−
A
)
−
1
in its simplest form.
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