Question

The system of equations
$x+y+z=5$
$x+2y+3z=9$
$x+3y+\lambda z=\mu$
has a unique solution, if

• A. $\lambda =5,\mu =13$
• B. $\lambda \ne 5$
• C. $\lambda =5,\mu \ne 13$
• D. $\mu \ne 13$

Answer 1

Answer: (B)

$\lambda \ne 5$

Given system of equations $AX=B$

where $A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& \lambda \end{array}\right]$, $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}5\\ 9\\ \mu \end{array}\right]$
For unique solution,

$D\ne 0$

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

$\Rightarrow (2\lambda -9)-(\lambda -3)+1\ne 0$

$\Rightarrow \lambda \ne 5$