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Global Maxima

If Ak=0 for...

Question

If $A^{k} = 0$ for some value of $k$ and $B = 1 + A + A^{2} + . . . + A^{k - 1},$ then $B^{- 1}$ equal

I - A

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I + A

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I - A^{k - 1}

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None of these

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Solution

The correct option is A $I - A$
Let $B = I + A + A^{2} + A^{3} + . . . + A^{k - 1}$
$∴ B (I - A) = (I + A + A^{2} + . . . + A^{k - 1}) (I - A)$
$= I - A + A - A^{2} + A^{2} + . . . + A^{k - 1} - A^{k} = I = A^{k} = I (∵ A^{k} = O)$
Hence, ${(I - A)}^{- 1} = I + A + A^{2} + . . . + A^{k} - 1 = B$
$\Rightarrow B^{- 1} = I - A$

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