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Question: If A is a nilpotent matrix of index 2, then find the value of \(A{(I + A)^n}\) for any positive inte...

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

Explanation

Solution

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.
A2=0{A^2} = 0
A3=0{A^3} = 0
A3=0....{A^3} = 0....
An=0{A^n} = 0
Now, we have to find the value of A(I+A)nA{(I + A)^n}
⇒A(I+A)n=A[nC0In+nC1In−1A+nC2In−2A2+....+nCnI0An]\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}]
⇒A(I+A)n=A[I+nA]\Rightarrow A{(I + A)^n} = A\left[ {I + nA} \right]
⇒A(I+A)n=AI+nA2\Rightarrow A{(I + A)^n} = AI + n{A^2}
⇒A(I+A)n=A\Rightarrow A{(I + A)^n} = A
∴\therefore The value of A(I+A)n=AA{(I + A)^n} = A

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