Question

The value of $\left|\begin{array}{ccc}{a}^2& 2ab& b^2\\ b^2& a^2& 2ab\\ 2ab& b^2& a^2\end{array}\right|$ is

• A. ${(a^2+b^2)}^3$
• B. ${(a^3+b^3)}^2$
• C. ${(a^4+b^4)}^2$
• D. ${(a^2+b^2)}^4$

Answer 1

Answer: (B)

Let $\Delta =\left|\begin{array}{ccc}{a}^2& 2ab& b^2\\ b^2& a^2& 2ab\\ 2ab& b^2& a^2\end{array}\right|$
Applying, $C_1\to C_2+C_3,$ we get $\Rightarrow$
$\Delta \left|\begin{array}{ccc}{(a+b)}^2& 2ab& b^2\\ {(a+b)}^2& a^2& 2ab\\ {(a+b)}^2& b^2& a^2\end{array}\right|$
$={(a+b)}^2\left|\begin{array}{ccc}1& 2ab& b^2\\ 1& a^2& 2ab\\ 1& b^2& a^2\end{array}\right|$
Apply $R_2\to R_1-R_1,R_3\Rightarrow R_3-R_1,$
we get $\Delta ={(a+b)}^2\left|\begin{array}{ccc}1& 2ab& b^2\\ 0& a^2-2ab& 2ab-b^2\\ 0& b^2-2ab& a^2-b^2\end{array}\right|$
$={(a+b)}^2\left|\begin{array}{cc}a(a-2b)& b(2a-b)\\ b(b-2a)& (a-b)(a+b)\end{array}\right|$
$={(a+b)}^2\{a(a-b)(a+b)(a-2b)$
$-b^2(b-2a)(2a-b)\}$
$={(a+b)}^2\{a(a^2-b^2)(a-2b)+b^2{(2a-b)}^2\}$
$={(a+b)}^2\{(a^2-b^2)(a^2-2ab)$
$+b^2(4a^2+b^2-4ab)\}$
$={(a+b)}^2\{a^4-a^2{b}^2-2a^{3}b+2a{b}^3$
$+4a^2{b}^2+b^4-4a{b}^3\}$
$={(a+b)}^2\{a^4+b^4+3a^2{b}^2-2a^{3}b-2a{b}^3\}$
$={(a+b)}^2\{{(a^2+b^2)}^2-2ab(a^2+b^2)+a^2{b}^2\}$
$={(a+b)}^2\{(a^2+b^2)(a^2+b^2-2ab)+a^2{b}^2\}$
$={(a+b)}^2\{(a^2+b^2){(a-b)}^2+a^2{b}^2\}$
$=(a^2+b^2)(a^2-b^2)+a^2{b}^2{(a+b)}^2$
$=(a^4-b^4)(a^2-b^2)+a^2{b}^2{(a+b)}^2$
$=a^6-a^2{b}^4-a^4{b}^2+b^6+a^4{b}^2+a^2{b}^4+2a^3{b}^3$
$=a^6+2a^3{b}^3+b^6={(a^3+b^3)}^2$