A) \[x=1,y=1,z=1\]
B) \[x=-1,y=1,z=1\]
C) \[x=1,y=-1,z=1\]
D) None of the above
Correct Answer: A
Solution :
In a given system of equations, \[A=\left[ \begin{matrix} 1 & 2 & 3 \\ 3 & -2 & 1 \\ 4 & 2 & 1 \\ \end{matrix} \right],B=\left[ \begin{align} & 6 \\ & 2 \\ & 7 \\ \end{align} \right],X=\left[ \begin{align} & x \\ & y \\ & z \\ \end{align} \right]\] Now, |A| = 1 (-2-2) - 2 (3-4) + 3 (6+ 8) = - 4 + 2 + 42 = 40 Again now adj\[(A)=\left[ \begin{matrix} -4 & 4 & 8 \\ 1 & -11 & 8 \\ 14 & 6 & -8 \\ \end{matrix} \right]\] \[\therefore \]\[{{A}^{-1}}=\frac{1}{|A|}adj(A)=\frac{1}{40}\left[ \begin{matrix} -4 & 4 & 8 \\ 1 & -11 & 8 \\ 14 & 6 & -8 \\ \end{matrix} \right]\] Now, \[X={{A}^{-1}}B=\frac{1}{40}\left[ \begin{matrix} -4 & 4 & 8 \\ 1 & -11 & 8 \\ 14 & 6 & -8 \\ \end{matrix} \right]\left[ \begin{align} & 6 \\ & 2 \\ & 7 \\ \end{align} \right]\] \[\Rightarrow \]\[\left[ \begin{align} & x \\ & y \\ & z \\ \end{align} \right]=\frac{1}{40}\left[ \begin{align} & 40 \\ & 40 \\ & 40 \\ \end{align} \right]\]\[\Rightarrow \]\[x=1,y=1,z=1\]
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