Question

Assertion :If a, b, c $ϵR$ and $a\ne b\ne c$ and x, y, z are non zero. Then the system of equations $ax+by+cz=0$
$bx+cy+az=0$
$cx+ay+bz=0$ has infinite solutions. Reason: If the homogeneous system of equations has non trivial solution, then it has infinitely many solutions.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Assertion:
Given system of equations can be written as
$AX=O$
Where O is zero matrix .
Here, $A=\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]$
Now, $D=\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$
$\Rightarrow D=3abc-a^3-b^3-c^3$
$\Rightarrow D=-(a^3+b^3+c^3-3abc)=0$
Clearly, $D_1=\left|\begin{array}{ccc}0& b& c\\ 0& c& a\\ 0& a& b\end{array}\right|=0$
Similarly, $D_2=D_3=0$
Hence, $D=D_1=D_2=D_3=0$
Hence, the system has infinite many solution.
Reason is also correct . A homogeneous system of eqn having non-trivial solution has infinite many solutions