Question

Show that each one of the following systems of linear equation is inconsistent:
4x − 2y = 3
6x − 3y = 5

Answer 1

The given system of equations can be expressed as follows:
\[AX = B\]
Here,
\[ A = \begin{bmatrix}4 & - 2 \\ 6 & - 3\end{bmatrix}, X = \binom{x}{y}\text{ and }B = \binom{3}{5}\]
\[ \left| A \right| = \begin{vmatrix}4 & - 2 \\ 6 & - 3\end{vmatrix}\]
\[ = \left( - 12 + 12 \right)\]
\[ = 0\]
\[ {\text{ Let }C}_{ij} {\text{ be the cofactors of the elements a }}_{ij}\text{ in }A =\left[ a_{ij} \right].\text{ Then,}\]
\[ C_{11} = - \left( 1 \right)^{1 + 1} \left( - 3 \right) = - 3, C_{12} = - \left( 1 \right)^{1 + 2} \left( 6 \right) = - 6\]
\[ C_{21} = - \left( 1 \right)^{2 + 1} \left( - 2 \right) = 2, C_{22} = - \left( 1 \right)^{2 + 2} \left( 4 \right) = 4\]
\[adj A = \begin{bmatrix}- 3 & - 6 \\ 2 & 4\end{bmatrix}^T \]
\[ = \begin{bmatrix}- 3 & 2 \\ - 6 & 4\end{bmatrix}\]
\[\left( adj A \right) B = \begin{bmatrix}- 3 & 2 \\ - 6 & 4\end{bmatrix}\binom{3}{5}\]
\[ = \binom{ - 9 + 10}{ - 18 + 20}\]
\[ = \binom{1}{2} \neq 0\]
Hence, the given system of equations is inconsistent.