Let S$_1$ and S$_2$ be respectively the sets of all $a \in \mathbb{R} - \{ 0\}$ for which the system of linear equations
$ax + 2ay - 3az = 1$
$(2a + 1)x + (2a + 3)y + (a + 1)z = 2$
$(3a + 5)x + (a + 5)y + (a + 2)z = 3$
has unique solution and infinitely many solutions. Then
Solution
Given system of equations
<br/><br/>
$$
\begin{aligned}
& a x+2 a y-3 a z=1 \\\\
& (2 a+1) x+(2 a+3) y+(a+1) z=2 \\\\
& (3 a+5) x+(a+5) y+(a+2) z=3 \\\\
& \text { Let } A=\left|\begin{array}{ccc}
a & 2 a & -3 a \\\\
2 a+1 & 2 a+3 & a+1 \\\\
3 a+5 & a+5 & a+2
\end{array}\right| \\\\
& =a\left|\begin{array}{ccc}
1 & 0 & 0 \\\\
2 a+1 & 1-2 a & 7 a+4 \\\\
3 a+5 & -5 a-5 & 10 a+17
\end{array}\right| \\\\
& =a\left(15 a^{2}+31 a+37\right) \\\\
& \text { Now } A=0 \\\\
& \Rightarrow a=0
\end{aligned}
$$
<br/><br/>
So, $S_{1}=R-\{0\}$ and at $a=0$
<br/><br/>
System has infinite solution but $a \in R-\{0\}$
<br/><br/>
$\therefore S_{2}=\Phi$
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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