행렬의 연산

$\begin{array}{cccccc} & & & & & \left\{\begin{array}{l}x+3y=4 \\ -2x-6y=3\end{array}\right. \\ \begin{array}{c}\text{Evaluate the determinant}\,D,\,\text{using the} \\ \text{coefficients of the variables.}\end{array} & & & & & D=|\begin{array}{cccc}1 & & & 3 \\ -2 & & & -6\end{array}| \\ & & & & & D=-6-(-6) \\ & & & & & D=0\end{array}$

We cannot use Cramer’s Rule to solve this system. But by looking at the value of the determinants ${D}_{x}$ and ${D}_{y},$ we can determine whether the system is dependent or inconsistent.

$\begin{array}{cccccc}\text{Evaluate the determinant}\,{D}_{x}. & & & & & \quad {D}_{x}=|\begin{array}{cccc}4 & & & 3 \\ 3 & & & -6\end{array}| \\ & & & & & \quad {D}_{x}=-24-9 \\ & & & & & \quad {D}_{x}=-33\end{array}$

Since all the determinants are not zero, the system is inconsistent. There is no solution.