Question

Find the value of the following determinants.
$\left|\begin{array}{ccc}-a\left(b^2+c^2-a^2\right)& 2b^3& 2c^3\\ 2a^3& -b\left(c^2+a^2-b^2\right)& 2c^3\\ 2a^3& 2b^3& -c{\left(a^2+b^2+c^2\right)}^2\end{array}\right|$

• A. $abc{\left(a^2+b^2+c^2\right)}^3$
• B. ${\left(a^2+b^2+c^2\right)}^2$
• C. $abc$
• D. None of these

Answer 1

Answer: (A)

Let $\Delta =\left|\begin{array}{ccc}-a\left(b^2+c^2-a^2\right)& 2b^3& 2c^3\\ 2a^3& -b\left(a^2+c^2-b^2\right)& 2c^3\\ 2a^3& 2b^3& -c\left(a^2+b^2-c^2\right)\end{array}\right|$

Taking $a,b,c$ common from $C_1$, and $C_3$ respectively, we get

$\Delta =abc\left|\begin{array}{ccc}{a}^2-b^2-c^2& 2b^2& 2c^2\\ 2a^2& b^2-c^2-a^2& 2c^2\\ 2a^2& 2b^2& c^2-a^2-b^2\end{array}\right|$
Applying $C_1\to C_1+C_2+C_3$, we get

$\Delta =abc\left|\begin{array}{ccc}{a}^2+b^2+c^2& 2b^2& 2c^2\\ a^2+b^2+c^2& b^2-c^2-a^2& 2c^2\\ a^2+b^2+c^2& 2b^2& c^2-a^2-b^2\end{array}\right|$

Taking $\left(a^2+b^2+c^2\right)$ common from $C_1$, we get

$\Delta =abc\left(a^2+b^2+c^2\right)\left|\begin{array}{ccc}1& 2b^2& 2c^2\\ 1& b^2-c^2-a^2& 2c^2\\ 1& 2b^2& c^2-a^2-b^2\end{array}\right|$

Applying $R_1\to R_1-R_2,R_2\to R_2-R_3$, we get

$\Delta =abc\left(a^2+b^2+c^2\right)\left|\begin{array}{ccc}0& b^2+c^2+a^2& 0\\ 0& -\left(b^2+c^2+a^2\right)& \left(a^2+b^2+c^2\right)\\ 1& 2b^2& c^2-a^2-b^2\end{array}\right|$

Taking $a^2+b^2+c^2$ common from $R_1$ and $R_2$, we get

$\Delta =abc{\left(a^2+b^2+c^2\right)}^3\left|\begin{array}{ccc}0& 1& 0\\ 0& -1& 1\\ 1& 2b^2& c^2-a^2-b^2\end{array}\right|$

Expanding along $C_1$ , we get

$\Delta =abc{\left(a^2+b^2+c^2\right)}^3\left(1\right)=abc{\left(a^2+b^2+c^2\right)}^3$