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If A^(3)=O, then prove that (I-A)^(-1) =...

If `A^(3)=O`, then prove that `(I-A)^(-1) =I+A+A^(2)`.

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Given`A^(3)=O`
Now, `(I-A) (I+A+A^(2))`
`=I^(2)+IA+IA^(2)-AI-A^(2)-A^(3)`
`=I+A+A^(2)-A-A^(2)-O`
`=I`
`:. (I-A)^(-1)=I+A+A^(2)`
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CENGAGE-MATRICES-Exercise 13.5
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  3. Find A^(-1) if A=|(0,1,1),(1,0,1),(1,1,0)| and show that A^(-1)=(A^(2)...

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  5. If A^(3)=O, then prove that (I-A)^(-1) =I+A+A^(2).

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  8. If A is a 2xx2 matrix such that A^(2)-4A+3I=O, then prove that (A+3I)^...

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  10. Prove that inverse of a skew-symmetric matrix (if it exists) is skew-s...

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  12. If A is a skew symmetric matrix, then B=(I-A)(I+A)^(-1) is (where I is...

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  13. Prove that ("adj. "A)^(-1)=("adj. "A^(-1)).

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  15. Show that the two matrices A, P^(-1) AP have the same characteristic r...

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