Question

The determinant $△=\left|\begin{array}{ccc}{a}^2(1+x)& ab& ac\\ ab& b^2(1+x)& bc\\ ac& bc& c^2(1+x)\end{array}\right|$ is divisible by

• A. $1+x$
• B. $(1+x{)}^2$
• C. $x^2$
• D. $x^2+1$

Answer 1

The correct option is A $1+x$

$△=\left|\begin{array}{ccc}{a}^2(1+x)& ab& ac\\ ab& b^2(1+x)& bc\\ ac& bc& c^2(1+x)\end{array}\right|$

Put $a=b=c=1$

$△=\left|\begin{array}{ccc}(1+x)& 1& 1\\ 1& (1+x)& 1\\ 1& 1& (1+x)\end{array}\right|$

We can see that it is divisible by $(1+x)$