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Question
The number of solutions of the system of equations:
2x + y − z = 7
x − 3y + 2z = 1
x + 4y − 3z = 5
Options
3
2
1
0
Solution
(d) 0
The given system of equations can be written in matrix form as follows:
\[ \begin{bmatrix}2 & 1 & - 1 \\ 1 & - 3 & 2 \\ 1 & 4 & - 3\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}7 \\ 1 \\ 5\end{bmatrix}\]
\[AX = B\]
Here,
\[ A = \begin{bmatrix}2 & 1 & - 1 \\ 1 & - 3 & 2 \\ 1 & 4 & - 3\end{bmatrix}, X = \begin{bmatrix}x \\ y \\ z\end{bmatrix}\text{ and }B = \begin{bmatrix}7 \\ 1 \\ 5\end{bmatrix}\]
Now,
\[\left| A \right|=2 \left( 9 - 8 \right) - 1\left( - 3 - 2 \right) - 1\left( 4 + 3 \right)\]
\[ = 2 + 5 - 7\]
\[ = 0\]
\[ {\text{ Let }C}_{ij} {\text{ be the cofactors of the elements a }}_{ij}\text{ in }A=\left[ a_{ij} \right].\text{ Then,}\]
\[ C_{11} = \left( - 1 \right)^{1 + 1} \begin{vmatrix}- 3 & 2 \\ 4 & - 3\end{vmatrix} = 1, C_{12} = \left( - 1 \right)^{1 + 2} \begin{vmatrix}1 & 2 \\ 1 & - 3\end{vmatrix} = 5, C_{13} = \left( - 1 \right)^{1 + 3} \begin{vmatrix}1 & - 3 \\ 1 & 4\end{vmatrix} = 7\]
\[ C_{21} = \left( - 1 \right)^{2 + 1} \begin{vmatrix}1 & - 1 \\ 4 & - 3\end{vmatrix} = - 1, C_{22} = \left( - 1 \right)^{2 + 2} \begin{vmatrix}2 & - 1 \\ 1 & - 3\end{vmatrix} = - 5, C_{23} = \left( - 1 \right)^{2 + 3} \begin{vmatrix}2 & 1 \\ 1 & 4\end{vmatrix} = - 7\]
\[ C_{31} = \left( - 1 \right)^{3 + 1} \begin{vmatrix}1 & - 1 \\ - 3 & 2\end{vmatrix} = 5, C_{32} = \left( - 1 \right)^{3 + 2} \begin{vmatrix}2 & - 1 \\ 1 & 2\end{vmatrix} = - 5, C_{33} = \left( - 1 \right)^{3 + 3} \begin{vmatrix}2 & 1 \\ 1 & - 3\end{vmatrix} = - 7\]
\[adj A = \begin{bmatrix}1 & 5 & 7 \\ - 1 & - 5 & - 7 \\ 5 & - 5 & - 7\end{bmatrix}^T = \begin{bmatrix}1 & - 1 & 5 \\ 5 & - 5 & - 5 \\ 7 & - 7 & - 7\end{bmatrix}\]
\[ \Rightarrow \left( adj A \right)B = \begin{bmatrix}1 & - 1 & 5 \\ 5 & - 5 & - 5 \\ 7 & - 7 & - 7\end{bmatrix}\begin{bmatrix}7 \\ 1 \\ 5\end{bmatrix}\]
\[ = \begin{bmatrix}7 - 1 + 25 \\ 35 - 5 - 25 \\ 49 - 7 - 35\end{bmatrix} = \begin{bmatrix}32 \\ 5 \\ 6\end{bmatrix}\neq 0\]
The given system of equations is inconsistent . Thus, it has no solution .
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