Question

If the system of the equations
$2x+ay+6z=8,x+2y+bz=5,x+y+3z=4$
has a unique solution, then which of the following is true regarding $a$ and $b$?

• A. $a=2,b=3$
• B. $a\ne 2,b=3$
• C. $a=2,b\ne 3$
• D. $a\ne 2,b\ne 3$

Answer 1

The correct option is D $a\ne 2,b\ne 3$

$\Delta =\left|\begin{array}{ccc}2& a& 6\\ 1& 2& b\\ 1& 1& 3\end{array}\right|$

$R_1\to R_1-2R_3,R_2\to R_2-R_3$
$\Delta =\left|\begin{array}{ccc}0& a-2& 0\\ 0& 1& b-3\\ 1& 1& 3\end{array}\right|$
$\Delta =(a-2)(b-3)$

$\Delta \ne 0$ for unique solution
$\therefore a\ne 2,b\ne 3$