Linear algebra adjoint proof question

In summary, the conversation discusses how to simplify the expression det (A^-1 det A) into (detA)^n and the use of properties of determinants. It is explained that this simplification is possible because of the property that for any number k and matrix M, det (kM) = k^n det(M).
  • #1
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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  • #2
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)
 

Related to Linear algebra adjoint proof question

1. What is the definition of a linear algebra adjoint?

The adjoint of a linear transformation T is the unique linear transformation T* that satisfies the property (T*(v), w) = (v, T(w)) for all vectors v and w in the vector space.

2. What is the importance of a linear algebra adjoint in mathematics?

The adjoint plays a crucial role in solving systems of linear equations and finding eigenvalues and eigenvectors of linear transformations. It also allows for the definition of the adjugate matrix, which is useful in finding the inverse of a matrix.

3. How is the adjoint of a linear transformation calculated?

To find the adjoint of a linear transformation, you can use the matrix representation of the transformation and take the transpose of the matrix. Alternatively, you can use the definition of the adjoint and solve for the adjoint transformation directly.

4. Can a linear algebra adjoint be undefined?

Yes, the adjoint of a linear transformation is undefined if the transformation is not square or if it is not invertible. Additionally, the adjoint may not exist if the vector space is not equipped with an inner product.

5. How is the adjoint of a linear algebra proof verified?

To verify a proof involving the adjoint of a linear transformation, you can use the definition of the adjoint to show that the property (T*(v), w) = (v, T(w)) holds for all vectors v and w in the vector space. You can also use the properties of transpose and inner product to simplify the proof.

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