1.18. Solve the system of equations using the matrix method

$$
\left\{\begin{array}{l}
x_{1}+2 x_{2}+4 x_{3}=5 \\
2 x_{1}+x_{2}+5 x_{3}=7 \\
3 x_{1}+2 x_{2}+6 x_{3}=9
\end{array}\right.
$$
Solution. Rewrite the system in matrix form: $A \cdot X=B$, where

$$
A=\left(\begin{array}{lll}
1 & 2 & 4 \\
2 & 1 & 5 \\
3 & 2 & 6
\end{array}\right), X=\left(\begin{array}{l}
x_{1} \\
x_{2} \\
x_{3}
\end{array}\right), B=\left(\begin{array}{l}
5 \\
7 \\
9
\end{array}\right)
$$

The solution is represented as $X=A^{-1} \cdot B$. The inverse matrix for the main matrix of the system is found in example 1.11, its form is

We have

$$
A^{-1}=\left(\begin{array}{ccc}
-\frac{2}{3} & -\frac{2}{3} & 1 \\
\frac{1}{2} & -1 & \frac{1}{2} \\
\frac{1}{6} & \frac{2}{3} & -\frac{1}{2}
\end{array}\right)
$$

$$
X=A^{-1} \cdot B=\left(\begin{array}{l}
1 \\
0 \\
1
\end{array}\right)
$$