Question

What is the determinant of a matrix to a power?

Answer 1

`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`.

Answer 2

`| 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`