Question

If $f\left(x\right)=$ $\left|\begin{array}{ccc}x+\lambda & x& x\\ x& x+\lambda & x\\ x& x& x+\lambda \end{array}\right|$ , then $f\left(3x\right)-f\left(x\right)=$

• A. $3x{\lambda }^2$
• B. $6x{\lambda }^2$
• C. $x{\lambda }^2$
• D. none of these

Answer 1

The correct option is B $6x{\lambda }^2$

$f\left(x\right)=\left|\begin{array}{ccc}x+\lambda & x& x\\ x& x+\lambda & x\\ x& x& x+\lambda \end{array}\right|$
$C_1\to C_1+C_2+C_3$
$=\left|\begin{array}{ccc}3x+\lambda & x& x\\ 3x+\lambda & x+\lambda & x\\ 3x+\lambda & x& x+\lambda \end{array}\right|$
$=(3x+\lambda )\left|\begin{array}{ccc}1& x& x\\ 1& x+\lambda & x\\ 1& x& x+\lambda \end{array}\right|$
$R_1\to R_1-R_2,R_2\to R_2-R_3$
$=(3x+\lambda )\left|\begin{array}{ccc}0& -\lambda & 0\\ 0& \lambda & -\lambda \\ 1& x& x+\lambda \end{array}\right|$
$\Rightarrow f\left(x\right)={\lambda }^2(3x+\lambda )$
Now, $f\left(3x\right)=\left|\begin{array}{ccc}3x+\lambda & 3x& 3x\\ 3x& 3x+\lambda & 3x\\ 3x& 3x& 3x+\lambda \end{array}\right|$
$C_1\to C_1+C_2+C_3$
$=\left|\begin{array}{ccc}9x+\lambda & 3x& 3x\\ 9x+\lambda & 3x+\lambda & 3x\\ 9x+\lambda & 3x& 3x+\lambda \end{array}\right|$
$=(9x+\lambda )\left|\begin{array}{ccc}1& 3x& 3x\\ 1& 3x+\lambda & 3x\\ 1& 3x& 3x+\lambda \end{array}\right|$
$=(9x+\lambda )\left|\begin{array}{ccc}0& -\lambda & 0\\ 0& \lambda & -\lambda \\ 1& 3x& 3x+\lambda \end{array}\right|$
$\Rightarrow f\left(3x\right)={\lambda }^2(9x+\lambda )$
Hence, $f\left(3x\right)-f\left(x\right)={\lambda }^{2}6x$