Question

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

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

Answer 1

Answer: (A)

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

Taking $a, b, c$ common from $C_{1}$, and $C_{3}$ respectively, we get

$\Delta=abc\left|\begin{matrix}a^{2}-b^{2}-c^{2}&2b^{2}&2c^{2}\\ 2a^{2}&b^{2}-c^{2}-a^{2}&2c^{2}\\ 2a^{2}&2b^{2}&c^{2}-a^{2}-b^{2}\end{matrix}\right|$
Applying $C_{1} \rightarrow C_{1} + C_{2} + C_{3}$, we get

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

Taking $\left(a^{2}+b^{2}+c^{2}\right)$ common from $C_{1}$, we get

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

Applying $R_{1} \rightarrow R_{1}-R_{2}, R_{2}\rightarrow R_{2}-R_{3}$, we get

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

Taking $a^{2} + b^{2} + c^{2}$ common from $R_{1}$ and $R_{2}$, we get

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

Expanding along $C_{1}$ , we get

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