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Question
Show that each one of the following systems of linear equation is inconsistent:
x + y − 2z = 5
x − 2y + z = −2
−2x + y + z = 4
Solution
The given system of equations can be written as follows:
AX = B
Here,
\[ A = \begin{bmatrix}1 & 1 & - 2 \\ 1 & - 2 & 1 \\ - 2 & 1 & 1\end{bmatrix}, X = \begin{bmatrix}x \\ y \\ z\end{bmatrix}\text{ and }B = \begin{bmatrix}5 \\ - 2 \\ 4\end{bmatrix}\]
\[\left| A \right| = \begin{vmatrix}1 & 1 & - 2 \\ 1 & - 2 & 1 \\ - 2 & 1 & 1\end{vmatrix}\]
\[ = 1\left( - 2 - 1 \right) - 1\left( 1 + 2 \right) - 2(1 - 4)\]
\[ = - 3 - 3 + 6\]
\[ = 0\]
\[ {\text{ Let }C}_{ij} {\text{ be the cofactors of the elements a }}_{ij}\text{ in }A\left[ a_{ij} \right]\text{. Then,}\]
\[ C_{11} = \left( - 1 \right)^{1 + 1} \begin{vmatrix}- 2 & 1 \\ 1 & 1\end{vmatrix} = - 3 , C_{12} = \left( - 1 \right)^{1 + 2} \begin{vmatrix}1 & 1 \\ - 2 & 1\end{vmatrix} = - 3, C_{13} = \left( - 1 \right)^{1 + 3} \begin{vmatrix}1 & - 2 \\ - 2 & 1\end{vmatrix} = - 3\]
\[ C_{21} = \left( - 1 \right)^{2 + 1} \begin{vmatrix}1 & - 2 \\ 1 & 1\end{vmatrix} = - 3, C_{22} = \left( - 1 \right)^{2 + 2} \begin{vmatrix}1 & - 2 \\ - 2 & 1\end{vmatrix} = - 3 , C_{23} = \left( - 1 \right)^{2 + 3} \begin{vmatrix}1 & 1 \\ - 2 & 1\end{vmatrix} = - 3\]
\[ C_{31} = \left( - 1 \right)^{3 + 1} \begin{vmatrix}1 & - 2 \\ - 2 & 1\end{vmatrix} = - 3, C_{32} = \left( - 1 \right)^{3 + 2} \begin{vmatrix}1 & - 2 \\ 1 & 1\end{vmatrix} = - 3, C_{33} = \left( - 1 \right)^{3 + 3} \begin{vmatrix}1 & 1 \\ 1 & - 2\end{vmatrix} = - 3\]
\[adj A = \begin{bmatrix}- 3 & - 3 & - 3 \\ - 3 & - 3 & - 3 \\ - 3 & - 3 & - 3\end{bmatrix}^T \]
\[ = \begin{bmatrix}- 3 & - 3 & - 3 \\ - 3 & - 3 & - 3 \\ - 3 & - 3 & - 3\end{bmatrix}\]
\[\left( adj A \right)B = \begin{bmatrix}- 3 & - 3 & - 3 \\ - 3 & - 3 & - 3 \\ - 3 & - 3 & - 3\end{bmatrix}\begin{bmatrix}5 \\ - 2 \\ 4\end{bmatrix}\]
\[ = \begin{bmatrix}- 15 + 6 - 12 \\ - 15 + 6 - 12 \\ - 15 + 6 - 12\end{bmatrix}\]
\[ = \begin{bmatrix}- 21 \\ - 21 \\ - 21\end{bmatrix} \neq 0\]
Hence, the given system of equations is consistent.
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