Question

If $△=\left|\begin{array}{ccc}a-b& b-c& c-a\\ b-c& c-a& a-b\\ c-a& a-b& b-c\end{array}\right|,$ then which of the following is correct?

• A. $△-2=0$
• B. $△-1=0$
• C. $△=0$
• D. $\Delta =abc$

Answer 1

The correct option is C $△=0$

Using 'Bagula' Rule :
$△=[a_1{b}_2{c}_3+b_1{c}_2{a}_3+c_1{a}_2{b}_3]-[a_3{b}_2{c}_1+b_3{c}_2{a}_1+c_3{a}_2{b}_1]$
$($sum of product of elements in principal diagonal and elements forming its 'star'] - [sum of product of elements in geometrical diagonal and elements forming its 'star'$)$
$\Rightarrow △=\left[\right(a-b\left)\right(b-c\left)\right(c-a)+(b-c\left)\right(c-a\left)\right(a-b)+(c-a\left)\right(a-b\left)\right(b-c\left)\right]-\left[\right(c-a{)}^3+(b-c{)}^3+(a-b{)}^3]\Rightarrow △=3(a-b\left)\right(b-c\left)\right(c-a)-[(c-a{)}^3+(b-c{)}^3+(a-b{)}^3]\Rightarrow △=0[\because (a-b)+(b-c)+(c-a)=0]$