Question

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 ______

Answer 1

Answer: 5

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