If $A = [\begin{matrix} 0 & 1 0 & 0 \end{matrix}]$ , $I$ is the unit matrix of order $2$ and $a, b$ are arbitrary constants then $(a I + b A)^{2}$ is equal to

a^{2} I + a b A

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a^{2} I + 2 a b A

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a^{2} I + b^{2} A

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None of these

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Solution

The correct option is A $a^{2} I + 2 a b A$
$A = [\begin{matrix} 0 & 1 0 & 0 \end{matrix}]$
$A^{2} = [\begin{matrix} 0 & 1 0 & 0 \end{matrix}] [\begin{matrix} 0 & 1 0 & 0 \end{matrix}]$
$\Rightarrow A^{2} = O$
Now, consider $(a I + b A)^{2} = (a I + b A) (a I + b A)$
$= a^{2} I + a b I A + b a A I + b^{2} A^{2}$
$= a^{2} I + 2 a b A (∵ A^{2} = O, a b = b a f o r r e a l n u m b e r s)$

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