Question

The existence of the unique solution of the system of equations
$x+y+z=\lambda$
$5x-y+\mu z=10$
$2x+3y-z=6$
depends on

• A. $\mu$ only
• B. $\lambda$ only
• C. $\lambda$ and $\mu$ both
• D. neither $\lambda$ nor $\mu$

Answer 1

The correct option is A $\mu$ only

Given system of equations $AX=B$
where $A=\left[\begin{array}{ccc}1& 1& 1\\ 5& -1& \mu \\ 2& 3& -1\end{array}\right]$, $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}\lambda \\ 10\\ 6\end{array}\right]$
Here, $|A|=23-\mu$
Clearly, here uniqueness of the system depends on $\mu$