- #1
SiddharthM
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Theorem: If A is a nilpotent square matrix (that is for some natural number k>0, A^k =0) then (I + A) is an invertible matrix.
Pf: Let B denote the inverse which will constructed directly. Let n be the smallest integer so that 2^n>k.
B=(1-A)(1+A^(2^1))(1+A^(2^2))...(1+A^(2^n))
then, B(1+A)=(1+A)B=1-A^(2^n)=1 - (A^(k))(A^(2n-k)) = 1
now if
QED
I wonder if this is a sufficient proof? I've never taken a linear algebra course so I really don't know!
Pf: Let B denote the inverse which will constructed directly. Let n be the smallest integer so that 2^n>k.
B=(1-A)(1+A^(2^1))(1+A^(2^2))...(1+A^(2^n))
then, B(1+A)=(1+A)B=1-A^(2^n)=1 - (A^(k))(A^(2n-k)) = 1
now if
QED
I wonder if this is a sufficient proof? I've never taken a linear algebra course so I really don't know!
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