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Hello Everyone :)

I have been facing a little difficulty when encountering such kind of problems . i have also written down my line of thinking and approach which i take to solve them. So, please try to give me the correct line of thinking while solving such problems:

1. If A is invertible , then prove that A + 2A^(-1) is invertible

My attempt : A + 2A^(-1) = A A A^-1 + 2A^(-1)

= (A^2 + 2I) A^(-1)

Now, A^(-1) is invertible , which means all i have to do is check for invertibility of

(A^2 + 2I)

What next . i mean i just don't get a clue after this. What is the correct line of thinking ?

2. If A^n = O , then Prove that A - I is invertible.

My attempt : i begin by assuming a matrix exists X so that i arrive back at the given condition

( A - I ) X = I

=> AX - IX = I

=> AX - ( I - A^n )X = I

=> AX - IX = I - A^nX

which solves nothing :(

little low on confidence :(

Thank you

I have been facing a little difficulty when encountering such kind of problems . i have also written down my line of thinking and approach which i take to solve them. So, please try to give me the correct line of thinking while solving such problems:

1. If A is invertible , then prove that A + 2A^(-1) is invertible

My attempt : A + 2A^(-1) = A A A^-1 + 2A^(-1)

= (A^2 + 2I) A^(-1)

Now, A^(-1) is invertible , which means all i have to do is check for invertibility of

(A^2 + 2I)

What next . i mean i just don't get a clue after this. What is the correct line of thinking ?

2. If A^n = O , then Prove that A - I is invertible.

My attempt : i begin by assuming a matrix exists X so that i arrive back at the given condition

( A - I ) X = I

=> AX - IX = I

=> AX - ( I - A^n )X = I

=> AX - IX = I - A^nX

which solves nothing :(

little low on confidence :(

Thank you

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