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Let $A = [\begin{array}{l} 1 \\ 1 \\ 1 \end{array}] and B = [\begin{array}{ccc} 9^{2} & - 10^{2} & 11^{2} \\ 12^{2} & 13^{2} & - 14^{2} \\ - 15^{2} & 16^{2} & 17^{2} \end{array}]$ , then the value of A'BA is:

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a

1224

b

1042

c

540

d

539

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detailed solution

Correct option is D

$\begin{array}{l} A^{'} BA = [\begin{array}{lll} 1 & 1 & 1 \end{array}] [\begin{array}{lcc} 9^{2} & - 10^{2} & 11^{2} \\ 12^{2} & 13^{2} & - 14^{2} \\ - 15^{2} & 16^{2} & 17^{2} \end{array}] [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] \\ = [\begin{array}{lll} 9^{2} + 12^{2} - 15^{2} & - 10^{2} + 13^{2} + 16^{2} & 11^{2} - 14^{2} + 17^{2} \end{array}] [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] \\ = [9^{2} + 12^{2} - 15^{2} - 10^{2} + 13^{2} + 16^{2} + 11^{2} - 14^{2} + 17^{2}] \\ = [539] \end{array}$

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