Question

In a square matrix A of order 3, the elements, $a_{i,i}$'s are the sum of the roots of the equation $x^2-(a+b)x+ab=0$; $a_{j,(j-1)}$'s are the all unity,$a_{j,(j+1)}$'s are product of the roots and rest of the elements are all
zero. Then value of the determinant is ___

• A. $(a^2+b^2)(a+b)$
• B. $0$
• C. $1$
• D. $a^2+b^2$

Answer 1

The correct option is A $(a^2+b^2)(a+b)$

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