Consider the following system of equations
$\alpha x+2y+z=1$
$2\alpha x+3y+z=1$
$3x+\alpha y+2z=\beta$
for some $\alpha,\beta\in \mathbb{R}$. Then which of the following is NOT correct.
Solution
$D=\left|\begin{array}{ccc}\alpha & 2 & 1 \\ 2 \alpha & 3 & 1 \\ 3 & \alpha & 2\end{array}\right|=0 \Rightarrow \alpha=-1,3$
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$D_{x}=\left|\begin{array}{ccc}2 & 1 & 1 \\ 3 & 1 & 1 \\ \alpha & 2 & \beta\end{array}\right|=0 \Rightarrow \beta=2$
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$D_{y}=\left|\begin{array}{ccc}\alpha & 1 & 1 \\ 2 \alpha & 1 & 1 \\ 3 & 2 & \beta\end{array}\right|=0$
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$D_{z}=\left|\begin{array}{ccc}\alpha & 2 & 1 \\ 2 \alpha & 3 & 1 \\ 3 & \alpha & \beta\end{array}\right|=0$
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$\beta=2, \alpha=-1$
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$\alpha=-1, \beta=2$ Infinite solution
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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