First slide

Applications of determinant

Question

$If A^{3} = 0, then I + A + A^{2} equals$

Moderate

A

$I - A$
B

${(I + A^{1})}^{- 1}$
C

${(I - A)}^{- 1}$
D

none of these

Solution

$Given A^{3} = 0$
$Now, (I - A) (I + A + A^{2})$
$\begin{array}{l} = I^{2} + I A + I A^{2} - A I - A^{2} - A^{3} \\ = I - A^{3} \\ = I \\ \Rightarrow (I - A)^{- 1} = I + A + A^{2} \end{array}$

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