Linear algebra adjoint proof question

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SUMMARY

The discussion focuses on understanding the transformation of the determinant expression det(A-1 det A) into (det A)n. The key point established is that det(A-1 det A) simplifies to (det A)n multiplied by det(A-1). This is derived from the property of determinants that states det(kM) = kn det(M) for any scalar k and matrix M. The confusion arises from the application of this property to the determinant of the product of matrices.

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  • Understanding of matrix operations and properties
  • Familiarity with determinants and their properties
  • Knowledge of linear algebra concepts, specifically adjoints
  • Basic understanding of matrix inverses
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  • Study the properties of determinants in detail
  • Learn about the adjoint of a matrix and its applications
  • Explore the relationship between matrix inverses and determinants
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Students studying linear algebra, particularly those tackling proofs involving determinants and matrix properties. This discussion is beneficial for anyone seeking clarity on the manipulation of determinant expressions in mathematical proofs.

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...

http://img28.imageshack.us/img28/8742/20130302114549.jpg

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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)
 

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