What is the determinant of a matrix to a power?

2 Answers
Daan J. · Truong-Son N.
Jul 20, 2015

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

Explanation:

A very important property of the determinant of a matrix, is that it 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 for two matrices,

#det(A^2)=det(A A)#

#=det(A)det(A)=det(A)^2#,

and for three matrices,

#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#.

Steve M
Dec 20, 2017

# | bb A^n | = | bb A|^n#

Explanation:

Using the property:

# |bbA bbB|=|bb A| \ |bb B| #

Then we have:

# | bb A^n | = |underbrace( bb A \ bb A \ bb A ... bb A)_("n terms") |#

# \ \ \ \ \ \ \ = | bb A| \ | bb A| \ | bb A| .... | bb A|#

# \ \ \ \ \ \ \ = | bb A|^n#

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