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QUESTION

# What is the determinant of a matrix to a power?

det(A^n)=det(A)^n

A very important property of the determinant of a matrix, is that is is a so called multiplicative function. It maps a matrix of numbers to a number in such a way that for two matrices A,B det(AB)=det(A)det(B). This means that det(A^2)=det(A A)=det(A)det(A)=det(A)^2, det(A^3)=det(A^2A)=det(A^2)det(A)=det(A)^2det(A)=det(A)^3 and so on.

Therefore in general det(A^n)=det(A)^n for any ninNN.