u8910520的部落格

When I preview the linear algebra.
In exercise,have a problem as following
A^2 is invertible <==> A^3 is invertible？
Try to consider general case.
A^k is invertible <==> A^m is invertible
where k,m belong to N
at first
consider A =>A^2
pf:we have AB=I=BA
consider AA(BB)=A(AB)B=AB=I
similarly to (BB)AA=I
then the proof is complete.
A=>A^n----(1) like this method.
it is easy to prove by math induction.
on the other hand.
A^2=>A
pf:we have A^2B=BA^2=I
A(AB)=I,so claim:(AB)A=I
ABA=ABAI=ABA(AAB)=A(BAA)AB=AAB=I
so we get A^2=>A
similarly method to porve A^3=>A^2
than we can get this conclusionA^(k+1)=>A^k
so when we have A^n is invertible
because A^n=>A^(n-1)、A^(n-1)=>A^(n-2)...A^2=>A
so we have a conclusion:A^n =>A^m ,n>m>=1---(2)
at last.consider A^m=>A^n n>m
pf:By(2)get A^m=>A,
by(1)get A=>A^n,
so A^m=>A^n QED.