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Question
Show that each one of the following systems of linear equation is inconsistent:
4x − 5y − 2z = 2
5x − 4y + 2z = −2
2x + 2y + 8z = −1
Solution
The given system of equations can be written as follows:
AX = B
Here,
\[ A = \begin{bmatrix}4 & - 5 & - 2 \\ 5 & - 4 & 2 \\ 2 & 2 & 8\end{bmatrix}, X = \begin{bmatrix}x \\ y \\ z\end{bmatrix}\text{ and }B = \begin{bmatrix}2 \\ - 2 \\ - 1\end{bmatrix}\]
\[\left| A \right| = \begin{vmatrix}4 & - 5 & - 2 \\ 5 & - 4 & 2 \\ 2 & 2 & 8\end{vmatrix}\]
\[ = 4\left( - 32 - 4 \right) + 5\left( 40 - 4 \right) - 2(10 + 8)\]
\[ = - 144 + 180 - 36\]
\[ = 0\]
\[ {\text{ Let }C}_{ij} {\text{ be the cofactors of the elements a }}_{ij}\text{ in }A\left[ a_{ij} \right].\text{ Then,}\]
\[ C_{11} = \left( - 1 \right)^{1 + 1} \begin{vmatrix}- 4 & 2 \\ 2 & 8\end{vmatrix} = 28, C_{12} = \left( - 1 \right)^{1 + 2} \begin{vmatrix}5 & 2 \\ 2 & 8\end{vmatrix} = - 36, C_{13} = \left( - 1 \right)^{1 + 3} \begin{vmatrix}5 & - 4 \\ 2 & 2\end{vmatrix} = 18\]
\[ C_{21} = \left( - 1 \right)^{2 + 1} \begin{vmatrix}- 5 & - 2 \\ 2 & 8\end{vmatrix} = 36 , C_{22} = \left( - 1 \right)^{2 + 2} \begin{vmatrix}4 & - 2 \\ 2 & 8\end{vmatrix} = 36 , C_{23} = \left( - 1 \right)^{2 + 3} \begin{vmatrix}4 & - 5 \\ 2 & 2\end{vmatrix} = - 18\]
\[ C_{31} = \left( - 1 \right)^{3 + 1} \begin{vmatrix}- 5 & - 2 \\ - 4 & 2\end{vmatrix} = - 18, C_{32} = \left( - 1 \right)^{3 + 2} \begin{vmatrix}4 & - 2 \\ 5 & 2\end{vmatrix} = - 18, C_{33} = \left( - 1 \right)^{3 + 3} \begin{vmatrix}4 & - 5 \\ 5 & - 4\end{vmatrix} = 9\]
\[adj A = \begin{bmatrix}28 & - 36 & 18 \\ 36 & 36 & - 18 \\ - 18 & - 18 & 9\end{bmatrix}^T \]
\[ = \begin{bmatrix}28 & 36 & - 18 \\ - 36 & 36 & - 18 \\ 18 & - 18 & 9\end{bmatrix}\]
\[\left( adj A \right)B = \begin{bmatrix}28 & 36 & - 18 \\ - 36 & 36 & - 18 \\ 18 & - 18 & 9\end{bmatrix}\begin{bmatrix}2 \\ - 2 \\ - 1\end{bmatrix}\]
\[ = \begin{bmatrix}56 - 72 + 18 \\ - 72 - 72 + 18 \\ 36 + 36 - 9\end{bmatrix}\]
\[ = \begin{bmatrix}2 \\ - 126 \\ 63\end{bmatrix} \neq 0\]
Hence, the given system of equations is consistent.
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