What is the determinant of a matrix to a power?
February 20th, 2023
##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##.
