The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 
$2\times 2$ matrix such that the trace of A is 3 and the trace of $A^3$ is $– 18,$ then the value of the determinant of A is _____

Let, $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],a+d=3$

$\begin{array}{l}\Rightarrow A^2=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}{a}^2+bc& ab+bd\\ ac+cd& bc+d^2\end{array}\right]\\ \Rightarrow A^3=\left[\begin{array}{cc}{a}^3+2abc+bcd& a^{2}b+abd+b^{2}c+b{d}^2\\ a^{2}c+acd+b{c}^2+c{d}^2& abc+2bcd+d^3\end{array}\right]\\ \Rightarrow a^3+d^3+3abc+3bcd=-18\\ \Rightarrow (a+d)(a^2+d^2-ad)+3bc(a+b)=-18\\ \Rightarrow (a+d)({(a+d)}^2-3ad)+3bc(a+d)=-18\\ \Rightarrow 3(9-3ad)+9bc=-18\\ \Rightarrow ad-bc=5\\ \Rightarrow \left|A\right|=5\end{array}$