Question

If A is a nilpotent matrix of index 2, then find the value of $A(I+A{)}^n$ for any positive integer n.
A. $A^{-1}$
B. $A$
C. $A^n$
D. $I_n$

Answer 1

Hint: Nilpotent matrix A, means some power of A is equal to the zero matrix.

Complete step-by-step answer:
Given A is a nilpotent matrix of index 2.
$A^2=0$
$A^3=0$
$A^3=0....$
$A^n=0$
Now, we have to find the value of $A(I+A{)}^n$
$\Rightarrow A(I+A{)}^n=A{[}^n{C}_0{I}^n{+}^n{C}_1{I}^{n-1}A{+}^n{C}_2{I}^{n-2}{A}^2+....{+}^n{C}_n{I}^0{A}^n]$
$\Rightarrow A(I+A{)}^n=A\left[I+nA\right]$
$\Rightarrow A(I+A{)}^n=AI+n{A}^2$
$\Rightarrow A(I+A{)}^n=A$
$\therefore$ The value of $A(I+A{)}^n=A$

Note: A nilpotent matrix is a square matrix N, such that $N^k=0,$ for some positive integer k. The smallest such k is sometimes called the index of N.