Question

The rank of the matrix $\left[\begin{array}{ccc}1& 2& 3\\ \lambda & 2& 4\\ 2& -3& 1\end{array}\right]$ is $3$ if

• A. $\lambda \ne \frac{18}{11}$
• B. $\lambda =\frac{18}{11}$
• C. $\lambda =-\frac{18}{11}$
• D. none of these

Answer 1

The correct option is A $\lambda \ne \frac{18}{11}$

$A=\left[\begin{array}{ccc}1& 2& 3\\ \lambda & 2& 4\\ 2& -3& 1\end{array}\right]$

Given, a rank of A =3=order of matrix

$|A|\ne 0$

$\Rightarrow \left|\begin{array}{ccc}1& 2& 3\\ \lambda & 2& 4\\ 2& -3& 1\end{array}\right|\ne 0$

$\Rightarrow -11\lambda +18\ne 0$

$\Rightarrow \lambda \ne \frac{18}{11}$