If
\[M=A_1A_2...A_n\]
then \[det(M)=det(A_1)det(A_2)...det(A_n)\]
where each matrix is an \[m \times m\]
matrix.Suppose then that an
\[m \times m\]
matrix satisfies \[A^2-3A+2I=0\]
. This factorises as \[(A-I)(A-2I)=0\]
then by the above property \[det(A-I)det(A-2I)=det(0)=0\]
.Hence
\[det(A-I)=0\]
or \[det(A-2I)=0\]
so \[det(A)=1\]
or \[det(A)=2^m\]
.