Which of the following is a nilpotent matrix?

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(C) matrix $A$ is called nilpotent if there exists a positive integer $m$ such that $A^m = 0$,where $0$ is the zero matrix.For option $C$,let $A = \be

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$\begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix}$

Vedclass · Answer

(C) matrix $A$ is called nilpotent if there exists a positive integer $m$ such that $A^m = 0$,where $0$ is the zero matrix.For option $C$,let $A = \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix}$.Then $A^2 = \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 	imes 0 + 0 	imes 1 & 0 	imes 0 + 0 	imes 0 \ 1 	imes 0 + 0 	imes 1 & 1 	imes 0 + 0 	imes 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$.Since $A^2 = 0$,the matrix $A$ is a nilpotent matrix of index $2$.

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