Question

For the system of equations:
x + 2y + 3z = 1
2x + y + 3z = 2
5x + 5y + 9z = 4

Options

• there is only one solution

• there exists infinitely many solution

• there is no solution

• none of these

Answer 1

(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.