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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
VerifiedVerified
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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.