Question

$\left|\begin{array}{ccc}{x}^2+2x& 2x+1& 1\\ 2x+1& x+2& 1\\ 3& 3& 1\end{array}\right|={(x-1)}^3.$

• A. True
• B. False

Answer 1

The correct option is A True

$\left|\begin{array}{ccc}{x}^2+2x& 2x+1& 1\\ 2x+1& x+2& 1\\ 3& 3& 1\end{array}\right|$

$R_1\to R_1-R_2{R}_3\to R_3-R_2$

we will get

$\left|\begin{array}{ccc}{x}^2-1& x-1& 0\\ 2x+1& x+2& 1\\ 2-2x& 1-x& 0\end{array}\right|=-1\left[\right(x^2-1\left)\right(1-x)-2(x-1\left)\right(1-x\left)\right]=(x-1)(x+1)(x-1)-2(x-1)(x-1)=(x-1{)}^2(x+1-2)=(x-1{)}^2(x-1)=(x-1{)}^3$