The index of the matrix A -Where A-beginbmatrix 2 - 411-1 -2lendbmatrix

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Standard IX

Mathematics

Nilpotent Matrix

The index of ...

Question

The index of the matrix A -
Where $A = [\begin{matrix} 2 & 4 - 1 & - 2 \end{matrix}]$ ___

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Solution

We know that index of a nilpotent matrix is the least positive integer m which satisfies $A^{m} = 0.$

Now all integers greater than this least positive integer will satisfy $A^{m} = 0.$

Since if $A^{m} = 0$

$A^{m + 1} = 0$

= $A^{m} \times A$

= 0 $\times$ A

= 0

So the least positive integer is the index.

In Question 2

$A^{1} \neq 0$

but $A^{2} = 0$

So 2 is that least positive integer which satisfies $A^{m} = 0.$ So index of A = 2.

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