Question

If the value of determinant $\left|\begin{array}{ccc}-16& 20& 36\\ 20& -25& 45\\ 36& 45& -81\end{array}\right|$ is $\Delta ,$ then $\sqrt{\Delta }$ equals to

• A. $180$
• B. $360$
• C. $720$
• D. $1440$

Answer 1

The correct option is B $360$

Given : $\Delta =\left|\begin{array}{ccc}-16& 20& 36\\ 20& -25& 45\\ 36& 45& -81\end{array}\right|$
Taking $\left(4\right)$ common from $R_1,\left(5\right)$ common from $R_2$ and $\left(9\right)$ common from $R_3$ respectively, we get
$\Delta =(4\cdot 5\cdot 9)\left|\begin{array}{ccc}-4& 5& 9\\ 4& -5& 9\\ 4& 5& -9\end{array}\right|$

Now taking $\left(4\right)$ common from $C_1,\left(5\right)$ common from $C_2$ and $\left(9\right)$ common from $C_3$ respectively, we get
$\Delta =(4\cdot 5\cdot 9{)}^2\left|\begin{array}{ccc}-1& 1& 1\\ 1& -1& 1\\ 1& 1& -1\end{array}\right|$

Now applying $R_1\to R_1+R_3,R_2\to R_2+R_3,$ we get
$\Delta =(180{)}^2\left|\begin{array}{ccc}0& 2& 0\\ 2& 0& 0\\ 1& 1& -1\end{array}\right|=(180{)}^2\times 4$
$\Rightarrow \sqrt{\Delta }=180\times 2=360$