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Matrix Equations and Determinants

Matrices have equations just like polynomials. Matrix equations may also factorise and one property transfers directly from the factorised form to determinant form.
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  
\[det(A)=2^m\]
.

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