Linear algebra adjoint proof question

baird.lindsay
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Homework Statement



I don't understand how line five counting from the top in the attached image. How does det (A^-1 det A) become (detA)^n? I get that the A^-1 was factored out but I don't get how (detA )= (detA)^n. Thank you...

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Homework Equations



Properties of determinates?

The Attempt at a Solution



This is the only part of the proof I don't get and i don't know what rule is det A det equal to...
 

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hi baird.lindsay! :smile:

(try using the X2 button just above the Reply box :wink:)
baird.lindsay said:
How does det (A^-1 det A) become (detA)^n?

it doesn't, it becomes (detA)n times det(A-1)

this is because, for any number k and for any matrix M, det (kM) = kndet(M) :wink:

(and detA is a number)
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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