## 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
$det(A)=2^m$
.