Question

# If ${A^3} = O$, then $I + A + {A^2}$ equalsA.$I - A$B.$\left( {I + {A^{ - 1}}} \right)$C.${\left( {I - A} \right)^{ - 1}}$D.None of these

Hint- As, it is given in the question ${A^3} = O$, and we can use this expression $I + A + {A^2}$, as a starting step, we can assume some value for this expression and multiply both side with A, as this will help us to make use of ${A^3} = O$, then the equation will be more simplified and we can use this equation in the main equation and find the value of the assume variable. Here, we should know one thing that when any matrix is multiplied by an identity matrix it always results in the same matrix.

Given, ${A^3} = O$, and we need to find the value of $I + A + {A^2}$.
Let, $y = I + A + {A^2}$ …….(1)
$Ay = A\left( {I + A + {A^2}} \right)$
$Ay = A + {A^2} + {A^3}$,
As the value of ${A^3} = O$, so substitute this value to the above equation.
$Ay = A + {A^2}$
$y = I + Ay \\ I = y - Ay \\ I = y\left( {1 - A} \right) \\ y = I{\left( {1 - A} \right)^{ - 1}} \\ y = {\left( {I - A} \right)^{ - 1}} \\$