Questions & Answers

Question

Answers

A.$I - A$

B.$\left( {I + {A^{ - 1}}} \right)$

C.${\left( {I - A} \right)^{ - 1}}$

D.None of these

Answer
Verified

Given, ${A^3} = O$, and we need to find the value of $I + A + {A^2}$.

Let, $y = I + A + {A^2}$ …….(1)

Multiply both sides with A.

$Ay = A\left( {I + A + {A^2}} \right)$

Now, simply the above equation,

$Ay = A + {A^2} + {A^3}$,

As the value of ${A^3} = O$, so substitute this value to the above equation.

$Ay = A + {A^2}$

Now, use the above equation in the equation (1).

$

y = I + Ay \\

I = y - Ay \\

I = y\left( {1 - A} \right) \\

y = I{\left( {1 - A} \right)^{ - 1}} \\

y = {\left( {I - A} \right)^{ - 1}} \\

$