Let A be a $3 \times 3$ real matrix such that $A (\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}) = (\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}); A (\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}) = (\begin{matrix} - 1 \\ 0 \\ 1 \end{matrix}) a n d$
$A (\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}) = (\begin{matrix} 1 \\ 1 \\ 2 \end{matrix}) i f X = {(x_{1} x_{2} x_{3})}^{T} a n d$
I is an identity matrix of order 3 then the system $(A - 3 I) X = [\begin{matrix} 1245 \\ 23 \\ 312 \end{matrix}]$ has

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no solution

unique solution

exactly two solutions

infinitely many solutions

answer is C.

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Detailed Solution

$L e t A = [\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}] \begin{array}{l} A [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] = [\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \end{matrix}] = [\begin{matrix} 1 \\ 1 \\ 2 \end{matrix}] \Rightarrow c_{1} = 1; c_{2} = 1; c_{3} = 2 \\ A [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}] = [\begin{matrix} c_{1} + a_{1} \\ c_{2} + a_{2} \\ c_{3} + a_{3} \end{matrix}] \Rightarrow a_{1} = - 2; a_{2} = - 1; a_{3} = - 1 \\ A [\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}] = [\begin{matrix} a_{1} + b_{1} \\ a_{2} + b_{2} \\ a_{3} + b_{3} \end{matrix}] = [\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}] \Rightarrow b_{1} = 3; b_{2} = 2; b_{3} = 1 \\ ∴ A = [\begin{matrix} - 2 & 3 & 1 \\ - 1 & 2 & 1 \\ - 1 & 1 & 2 \end{matrix}] \\ \Rightarrow | A - 3 I | = | \begin{matrix} - 5 & 3 & 1 \\ - 1 & - 1 & 1 \\ - 1 & 1 & - 1 \end{matrix} | \neq 0 \end{array}$
$∴$ Given system of equations has a unique solution