If A is a nilpotent matrix of index 2,then A( | vedclass

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(D) Given that $A$ is a nilpotent matrix of index $2$,we have $A^{2} = O$,where $O$ is the null matrix.Since $A^{2} = O$,it follows that $A^{n} = O$ f

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$A$

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(D) Given that $A$ is a nilpotent matrix of index $2$,we have $A^{2} = O$,where $O$ is the null matrix.Since $A^{2} = O$,it follows that $A^{n} = O$ for all $n \geq 2$.Using the binomial expansion for $(I_2+A)^{51}$:$(I_2+A)^{51} = \binom{51}{0} I_2^{51} + \binom{51}{1} I_2^{50} A + \binom{51}{2} I_2^{49} A^{2} + \dots + \binom{51}{51} A^{51}$.Since $A^{2} = O$,all terms containing $A^{k}$ for $k \geq 2$ become $O$.Thus,$(I_2+A)^{51} = I_2 + 51A$.Now,multiply by $A$:$A(I_2+A)^{51} = A(I_2 + 51A) = AI_2 + 51A^{2}$.Since $AI_2 = A$ and $A^{2} = O$,we get:$A(I_2+A)^{51} = A + 51(O) = A$.

If $A$ is a nilpotent matrix of index $2$,then $A(I_2+A)^{51}$ is equal to (where $I_2$ is the identity matrix of order $2$)

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