What is the determinant of a matrix to a power?

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What is the determinant of a matrix to a power?

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Answer 1 · 2015-07-20T01:07:30

$det (A^{n}) = {det (A)}^{n}$ $det(A^n)=det(A)^n$

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 $A, B$ $A,B$ ,

$det (A B) = det (A) det (B)$ $det(AB)=det(A)det(B)$ .

This means that for two matrices,

$det (A^{2}) = det (A A)$ $det(A^2)=det(A A)$

$= det (A) det (A) = {det (A)}^{2}$ $=det(A)det(A)=det(A)^2$ ,

and for three matrices,

$det (A^{3}) = det (A^{2} A)$ $det(A^3)=det(A^2A)$

$= det (A^{2}) det (A)$ $=det(A^2)det(A)$

$= {det (A)}^{2} det (A)$ $=det(A)^2det(A)$

$= {det (A)}^{3}$ $=det(A)^3$

and so on.

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

Answer 2 · 2017-12-20T02:47:51

$| bb A^n | = | bb A|^n$

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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What is the determinant of a matrix to a power?

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