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Properties of Matrix Multiplication

If A=[ ab b2...

Question

If $A = [\begin{matrix} a b & b^{2} - a^{2} & - a b \end{matrix}],$ then $A$ is

A

Involutory matix

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B

Idempotent matrix

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C

Nilpotent matrix of index

2

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D

none of these

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Solution

The correct option is C Nilpotent matrix of index $2$
$A^{2} = [\begin{matrix} a b & b^{2} - a^{2} & - a b \end{matrix}] [\begin{matrix} a b & b^{2} - a^{2} & - a b \end{matrix}] = [\begin{matrix} 0 & 0 0 & 0 \end{matrix}]$

$A^{2} = O \Rightarrow A$ is Nilpotent matrix having index 2

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Properties of Matrix Multiplication

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