Question

If $\left|\begin{array}{ccc}{\lambda }^2+3\lambda & \lambda -1& \lambda +3\\ \lambda +1& 1-2\lambda & \lambda -4\\ \lambda -2& \lambda +4& 3\lambda \end{array}\right|=p{\lambda }^4+q{\lambda }^3+r{\lambda }^2+s\lambda +t$ be an identity in $\lambda$, where p, q, r, s and t are constants, find the value of t.

Answer 1

Given $\left|\begin{array}{ccc}{\lambda }^2+3\lambda & \lambda -1& \lambda +3\\ \lambda +1& 1-2\lambda & \lambda -4\\ \lambda -2& \lambda +4& 3\lambda \end{array}\right|=p{\lambda }^4+q{\lambda }^3+r{\lambda }^2+s\lambda +t$
Putting $\lambda =0$ in given identity.

$\left|\begin{array}{ccc}0& -1& 3\\ 1& 1& -4\\ -2& 4& 0\end{array}\right|=0+0+0+0+t$

$\Rightarrow t=10$