Question

The determinant $D=\left|\begin{array}{ccc}1& x& x^2\\ x^2& 1& x\\ x& x^2& 1\end{array}\right|$ is equal to

• A. ${\left(1-x^3\right)}^2$
• B. ${\left(1-x^2\right)}^3$
• C. ${\left(x-x^2\right)}^3$
• D. none

Answer 1

Answer: (A)

Using $C_1\to C_1+C_2+C_3$
$D=\left(1+x+x^2\right)\left|\begin{array}{ccc}1& x& x^2\\ 1& 1& x\\ 1& x^2& 1\end{array}\right|=\left(x^2+x+1\right)\left|\begin{array}{ccc}0& x-1& x(x-1)\\ 0& -\left(x^2-1\right)& x-1\\ 1& x^2& 1\end{array}\right|$
$=(x-1{)}^2\left(x^2+x+1\right)\left|\begin{array}{ccc}0& 1& x\\ 0& -(x+1)& 1\\ 1& x^2& x\end{array}\right|=(x-1{)}^2\left(x^2+x+1\right)[1+x(x+1)]$
$={\left(x^2+x+1\right)}^2(x-1{)}^2={\left(x^3-1\right)}^2$