The trace of a square matrix is defined to be the sum of its...

Let A = \left[\begin{cases}a & b \\ c & 3 - a\end{cases}\right] ⇒ \left|A\right| = 3a - {a}^{2} - bc

{A}^{3} = \left[\begin{cases}a & b \\ c & 3 - a\end{cases}\right]\left[\begin{cases}a & b \\ c & 3 - a\end{cases}\right]\left[\begin{cases}a & b \\ c & 3 - a\end{cases}\right]

= \left[\begin{cases}{a}^{2} + bc & ab + 3b - ab \\ ac + 3c - ac & cb + ( 3 - a{)}^{2}\end{cases}\right]\left[\begin{cases}a & b \\ c & 3 - a\end{cases}\right]

= \left[\begin{cases}{a}^{2} + bc & 3b \\ 3c & cb + ( 3 - a{)}^{2}\end{cases}\right]\left[\begin{cases}a & b \\ c & 3 - a\end{cases}\right]

$tr \left({A}^{3}\right) = {a}^{3} + abc + 3bc + 3bc + 3bc - abc + ( 3 - a{)}^{3} = - 18$

${a}^{3} + 9bc + {\left(3 - a\right)}^{3} = - 18$

$9bc + 27 - 9a ( 3 - a ) = - 18$

$bc + 3 - 3a + {a}^{2} = - 2$

$3a - {a}^{2} - bc = 5 ⇒ | A | = 5$