Question

Using the properties of determinants, solve the following for x:
$\left[\begin{array}{ccc}X+2& X+6& X-1\\ X+6& X-1& X+2\\ X-1& X+2& X+6\end{array}\right]=0$

Answer 1

$\left[\begin{array}{ccc}X+2& X+6& X-1\\ X+6& X-1& X+2\\ X-1& X+2& X+6\end{array}\right]=0$

$R_1\to R_1-R_3$

$R_2\to R_2-R_3$

$\left[\begin{array}{ccc}3& 4& -7\\ 7& -3& -4\\ X-7& X+2& X+6\end{array}\right]=0$

$3[-3X-18+4X+8]-4[7X+42+4X-28]-7[7X+14+3X-21]=0$

$3[X-10]-4[11X+14]-7[10X-7]=0$

$=3X-30-44X-56-70X+49=0$

$=111X=86-49$

$=111X=37$

$\Rightarrow X=\frac{37}{111}$