Question

Investigate for what values of $\lambda ,\mu$ the simultaneous equations $x+y+z=6$, $x+2y+3z=10$, $x+2y+\lambda z=\mu$ have
(i) no solution (ii) a unique solution and (iii) an infinite number of solutions.

Answer 1

The given system can be written as

$\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 2& \lambda \end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}6\\ 10\\ \mu \end{array}\right]$

$AX=B$

The augmented matrix.

$(A,B)=\left[\begin{array}{cccc}1& 1& 1& 6\\ 1& 2& 3& 10\\ 1& 2& \lambda & \mu \end{array}\right]$

$=\left[\begin{array}{cccc}1& 1& 1& 6\\ 1& 2& 3& 4\\ 1& 2& \lambda -3& \mu -10\end{array}\right]\begin{array}{c}\\ R_2\to R_2-R_1\\ R_3\to R_3-R_2\end{array}$

Case (i) : $\lambda -3=0$ and $\mu -10\ne 0$

(ie) $\lambda =3,\mu \ne 10$

$\rho \left(A\right)=2,\rho (A,B)=3$

$\therefore \rho \left(A\right)\ne \rho (A,B)$

The given system is inconsistent but has no solution.

Case (ii) : $\lambda -3\ne 0$ and $\mu \in R$

(ie) $\lambda \ne 3$

\rho(A) = \rho(A,B) =3$

The given system is consistent and has unique solution.

Case (iii) : $\lambda =3$ and $\mu =10$

$\rho \left(A\right)=\rho (A,B)=2<$ number of unknowns.

The given system is consistent but has an infinite number of solutions.