1.

Let A and B be two matrices different from identify matrix such that AB=BA and A^(n)-B^(n) is invertible for some positive integer 'n'. IfA^(n)-B^(n)=A^(n+1)-B^(n+1)=A^(n+2)-B^(n+2), then

Answer»

I-A is non singular
I-B is non singular
I-A is singular
I-B is singular

SOLUTION :`A^(n+2)-B^(n+2)=(A+B)(A^(n+1)-B^(n+1))-AB(A^(n)-B^(n))`
`impliesA^(n)-B^(n)=(A+B)(A^(n)-B^(n))-AB(A^(n)-B^(n))`
`impliesI=A+B-AB""[becauseA^(n)-B^(n)"is invertible"]`
`implies(I-A)(I-B)=0`
As A, `B ne l`, we GET
I-A and I-B are singular matrices.


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