Question

$\left|\begin{array}{ccc}1+x& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+x\end{array}\right|$ is equal to

• A. $x^2(x+3)$
• B. $3x^3$
• C. $0$
• D. $x^3$

Answer 1

Answer: (A)

$x^2(x+3)$

$\left|\begin{array}{ccc}1+x& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+x\end{array}\right|$
$C_1\to C_1+C_2+C_3$
$\left|\begin{array}{ccc}x+3& 1& 1\\ x+3& 1+x& 1\\ x+3& 1& 1+x\end{array}\right|$
$=(x+3)\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+x\end{array}\right|$
$C_1\to C_1-C_2$
$=(x+3)\left|\begin{array}{ccc}0& 1& 1\\ 0& 1+x& 1\\ -x& 1& 1+x\end{array}\right|$
On expanding we get,
$=x^2(x+3)$