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Let A=\begin{bmatrix} 1&1 & 3\\ 5 &2 &6 \\ -2& -1 & -3 \end{bmatrix}. The A is

Option: 1

Nilpotent


Option: 2

Idempotent


Option: 3

Scalar


Option: 4

None of these 


Answers (1)

best_answer

As we have learned

Nilpotent matrix -

A^{m}=O

- wherein

m is the least positive integer and m is called the index

 

 

A^2=\begin{bmatrix} 0 &0 &0 \\ 3&3 &9 \\ -1& -1 & -3 \end{bmatrix}\neq A

            Hence not Idempotent

            A^3=A^2A=\begin{bmatrix} 0 & 0&0 \\ 3& 3 &9 \\ -1&-1 &-3 \end{bmatrix}\begin{bmatrix} 1 &1 &3 \\ 5& 2& 6\\ -2& -1 &-3 \end{bmatrix}

                  =\begin{bmatrix} 0 & 0 & 0\\ 0& 0& 0\\ 0&0 & 0 \end{bmatrix}=0

            Here A is nilpotent matrix of Index 3.

 

Posted by

Ritika Harsh

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