Let $Λ = [\begin{array}{lll} 1 0 0 \\ 2 1 0 \\ 3 2 1 \end{array}]$ if u₁ and u₂ are column matrices such that $A u_{1} = [\begin{array}{l} 1 \\ 0 \\ 0 \end{array}]$ and $A u_{2} = [\begin{array}{l} 0 \\ 1 \\ 0 \end{array}]$ then $u_{1} + u_{2}$ equal to

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$[\begin{array}{r} 1 \\ 1 \\ - 1 \end{array}]$

$[\begin{array}{r} 1 \\ - 1 \\ - 1 \end{array}]$

$[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}]$

$[\begin{array}{r} 1 \\ - 1 \\ 0 \end{array}]$

answer is D.

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Detailed Solution

$A (u_{1} + u_{2}) = [\begin{array}{l} 1 \\ 1 \\ 0 \end{array}]; | A | = 1, But A^{- 1} = \frac{1}{| A |} (adj A)$
$\Rightarrow u_{1} + u_{2} = A^{- 1} [\begin{array}{l} 1 \\ 1 \\ 0 \end{array}] = [\begin{array}{ccc} 1 & 0 & 0 \\ - 2 & 1 & 0 \\ 1 & - 2 & 1 \end{array}] [\begin{array}{l} 1 \\ 1 \\ 0 \end{array}] = [\begin{array}{r} 1 \\ - 1 \\ - 1 \end{array}]$