Question

If $Ifp{\lambda }^4+q{\lambda }^3+r{\lambda }^2+s\lambda +t=\left|\begin{array}{ccc}{\lambda }^2+3\lambda & \lambda -1& \lambda +3\\ \lambda +1& 2-\lambda & \lambda -4\\ \lambda -3& \lambda +4& 3\lambda \end{array}\right|$ then the value of $t$ is

Answer 1

Answer: 18

Since it is an indentity in$\lambda$ so it is satisfied by every value of $\lambda$. Now put $\lambda =0$ in the given equation, we have
$t=\left|\begin{array}{ccc}0& -1& 3\\ 1& 2& -4\\ -3& 4& 0\end{array}\right|=12+30=18$