Question

Using matrix method, find the values of $\lambda$ and $\mu$ so that the system of equations
$2x-3y+5z=12$
$3x+y+\lambda z=\mu$
$x-7y+8z=17$
has infinite solutions , find $\lambda +\mu$

Answer 1

Augumented matrix is $\left[\begin{array}{cccc}2& -3& 5& 12\\ 3& 1& \lambda & \mu \\ 1& -7& 8& 17\end{array}\right]$
applying $R_1\to R_1-R_3$
$\left[\begin{array}{cccc}1& 4& -3& -5\\ 3& 1& \lambda & \mu \\ 1& -7& 8& 17\end{array}\right]$
applying $R_3\to R_3-R_1$ and $R_2\to R_2-3R_1$
$\left[\begin{array}{cccc}1& 4& -3& -5\\ 0& -11& \lambda +9& \mu +15\\ 0& -11& 11& 22\end{array}\right]$
given system has infinite solutions if $\lambda +9=11$ and $\mu +15=22$
$\Rightarrow \lambda =2$ and $\mu =7$
$\therefore \lambda +\mu =9$