## 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 \[det(A)=2^m\]

.