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प्रश्न
For the system of equations:
x + 2y + 3z = 1
2x + y + 3z = 2
5x + 5y + 9z = 4
विकल्प
there is only one solution
there exists infinitely many solution
there is no solution
none of these
उत्तर
(a) there is only one solution
The given system of equations can be written in matrix form as follows:
\[\begin{bmatrix}1 & 2 & 3 \\ 2 & 1 & 3 \\ 5 & 5 & 9\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}1 \\ 2 \\ 4\end{bmatrix}\]
Here,
\[A=\begin{bmatrix}1 & 2 & 3 \\ 2 & 1 & 3 \\ 5 & 5 & 9\end{bmatrix},X=\begin{bmatrix}x \\ y \\ z\end{bmatrix}\text{ and }B = \begin{bmatrix}1 \\ 2 \\ 4\end{bmatrix}\]
Now,
\[\left| A \right| = \begin{vmatrix}1 & 2 & 3 \\ 2 & 1 & 3 \\ 5 & 5 & 9\end{vmatrix}\]
\[ = 1\left( 9 - 15 \right) - 2\left( 18 - 15 \right) + 3\left( 10 - 5 \right)\]
\[ = - 6 - 6 + 15\]
\[ = 3 \neq 0\]
\[ \Rightarrow \left| A \right|\neq 0 \]
So, the given system of equations has a unique solution.
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