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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

Gunjita

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