Question

If $A$ is a nilpotent matrix of index $2$, then for any positive integer $n$, $A(I+A{)}^n$ is equal to

• A. $A^{-1}$
• B. $A$
• C. $A^n$
• D. $I_n$

Answer 1

Answer: (B)

$A$

Since $A$ is a nilpotent matrix of index $2$

$A^2=0$
$A^3=0$
$A^4=0$ ....
$A^n=0$
Now, $A(I+A{)}^n=A{[}^n{C}_0{I}^n{+}^n{C}_1{I}^{n-1}A{+}^n{C}_2{I}^{n-2}{A}^2+.....{+}^n{C}_n{I}^0{A}^n]$

$=A[I+nA]$

$=AI+n{A}^2$

$=A$

$\Rightarrow A(I+A{)}^n=A$