Question

Prove that $\left|\begin{array}{ccc}{a}^2+2a& 2a+1& 1\\ 2a+1& a+2& 1\\ 3& 3& 1\end{array}\right|$ = 0
According as a = 1

Answer 1

Apply $R_1-R_2$ and $R_2-R_3$ thus making two zeros.

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

or $△=(a-1{)}^2\left|\begin{array}{cc}a+1& 1\\ 2& 1\end{array}\right|=(a-1{)}^3$

Clearly $△$ = 0 according as a = 1.