The number of values of $\alpha$ for which the system of equations :
x + y + z = $\alpha$
$\alpha$x + 2$\alpha$y + 3z = $-$1
x + 3$\alpha$y + 5z = 4
is inconsistent, is
Solution
$\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ \alpha & 2 \alpha & 3 \\ 1 & 3 \alpha & 5\end{array}\right|$
<br/><br/>
$$
\begin{aligned}
&=1(10 \alpha-9 \alpha)-1(5 \alpha-3)+1\left(3 \alpha^{2}-2 \alpha\right) \\\\
&=\alpha-5 \alpha+3+3 \alpha^{2}-2 \alpha \\\\
&=3 \alpha^{2}-6 \alpha+3
\end{aligned}
$$
<br/><br/>
For inconsistency $\Delta=0$ i.e. $\alpha=1$
<br/><br/>
Now check for $\alpha=1$
<br/><br/>
$x+y+z=1\quad\quad...(i)$
<br/><br/>
$x+2 y+3 z=-1\quad\quad...(ii)$
<br/><br/>
$x+3 y+5 z=4\quad\quad...(iii)$
<br/><br/>
By (ii) $\times 2-$ (i) $\times 1$
<br/><br/>
$x+3 y+5 z=-3$
<br/><br/>
so equations are inconsistent for $\alpha=1$
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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