Question

Solve the equations:
(1) $\left|\begin{array}{ccc}a& a& x\\ m& m& m\\ b& x& b\end{array}\right|=0$
(2) $\left|\begin{array}{ccc}15-2x& 11& 10\\ 11-3x& 17& 16\\ 7-x& 14& 13\end{array}\right|=0$.

Answer 1

$\left|\begin{array}{ccc}a& a& x\\ m& m& m\\ b& x& b\end{array}\right|=0C_1=C_1-C_2\Rightarrow \left|\begin{array}{ccc}0& a& x\\ 0& m& m\\ b-x& x& b\end{array}\right|=0$

Expanding along $C_1$

$\Rightarrow 0+0+(b-x)\{am-mx\}=0\Rightarrow (b-x)m(a-x)=0\Rightarrow (x-a)(x-b)=0\Rightarrow x=a,b$

(2) $\left|\begin{array}{ccc}15-2x& 11& 10\\ 11-3x& 17& 16\\ 7-x& 14& 13\end{array}\right|=0$

$R_1=R_1{+R}_2-2R_3\Rightarrow \left|\begin{array}{ccc}12-3x& 0& 0\\ 11-3x& 17& 16\\ 7-x& 14& 13\end{array}\right|=0$

Expanding along $R_1$

$\Rightarrow (12-3x)\{17\times 13-16\times 14\}-0+0\Rightarrow (12-3x)(-3)=0\Rightarrow x=4$