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

Let $A=\begin{bmatrix}a & b \\ c & d\end{bmatrix}, a+d=3$
$\Rightarrow A^{2}=\begin{bmatrix}a & b \\ c & d\end{bmatrix} \begin{bmatrix}a & b \\ c & d\end{bmatrix}=\begin{bmatrix}a^{2}+b c & a b+b d \\ a c+c d & b c+d^{2}\end{bmatrix}$
$\Rightarrow A^{3}=\begin{bmatrix}a^{3}+2 a b c+b c d & a^{2} b+a b d+b^{2} c+b d^{2} \\ a^{2} c+a c d+b c^{2}+c d^{2} & a b c+2 b c d+d^{3}\end{bmatrix}$
$\Rightarrow a^{3}+d^{3}+3 a b c+3 b c d=-18$
$\Rightarrow (a+d)\left(a^{2}+d^{2}-a d\right)+3 b c(a+d)=-18$
$\Rightarrow (a+d)\left((a+d)^{2}-3 a d\right)+3 b c(a+d)=-18$
$\Rightarrow 3(9-3 a d)+9 b c=-18$
$\Rightarrow a d-b c=5$
$\Rightarrow |A|=5$