Let the system of linear equations
$x + 2y + z = 2$,
$\alpha x + 3y - z = \alpha$,
$- \alpha x + y + 2z = - \alpha$
be inconsistent. Then $\alpha$ is equal to :
Solution
<p>$x + 2y + z = 2$</p>
<p>$\alpha x + 3y - z = \alpha$</p>
<p>$- \alpha x + y + 2z = - \alpha$</p>
<p>$$\Delta = \left| {\matrix{
1 & 2 & 1 \cr
\alpha & 3 & { - 1} \cr
{ - \alpha } & 1 & 2 \cr
} } \right| = 1(6 + 1) - 2(2\alpha - \alpha ) + 1(\alpha + 3\alpha )$$</p>
<p>$= 7 + 2\alpha$</p>
<p>$\Delta = 0 \Rightarrow \alpha = - {7 \over 2}$</p>
<p>$${\Delta _1} = \left| {\matrix{
2 & 2 & 1 \cr
\alpha & 3 & { - 1} \cr
{ - \alpha } & 1 & 2 \cr
} } \right| = 14 + 2\alpha \ne 0$$ for $\alpha = - {7 \over 2}$</p>
<p>$\therefore$ For no solution $\alpha = - {7 \over 2}$</p>
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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