Question

Assertion :If the system of equations $\lambda x+(b-a)y+(c-a)z=0,(a-b)x+\lambda y+(c-b)z=0,(a-c)x+(b-c)y+\lambda z=0$ has a non-trivial solution, then the value of $\lambda$ is 0. Reason: The value of skew-symmetric matrix of order 3 is zero.

• 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

For non-trivial solution,
$\left|\begin{array}{ccc}\lambda & b-a& c-a\\ a-b& \lambda & c-b\\ a-c& b-c& \lambda \end{array}\right|=0$
$\Rightarrow \lambda ({\lambda }^2+b^2+c^2-2bc)+\lambda {(a-b)}^2+\lambda {(a-c)}^2=0$
$\lambda [{\lambda }^2+b^2+c^2-2bc+a^2+b^2-2ab+a^2+c^2-2ac]=0$
$\Rightarrow \lambda =0$
We know that the determinant value of skew-symmetric matrix of odd order is 0.