How to Expand a Direct Determinant in Homework?

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SUMMARY

The discussion focuses on expanding the determinant of the matrix expression det(I + A dt) using properties of matrix exponentials and determinants. The solution involves rewriting I + A dt as exp(A dt) and applying the determinant property det(exp(A)) = exp(tr(A)). This leads to the conclusion that for small dt, det(I + A dt) approximates to 1 + tr(A) dt, confirming the relationship through both exponential and determinant formulas.

PREREQUISITES
  • Matrix theory, specifically properties of determinants
  • Understanding of matrix exponentials
  • Familiarity with the trace function in linear algebra
  • Basic calculus, particularly limits and small perturbations
NEXT STEPS
  • Study the properties of matrix exponentials in detail
  • Learn about the trace function and its applications in linear algebra
  • Explore advanced determinant formulas and their derivations
  • Investigate perturbation theory in linear algebra
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Students in linear algebra, mathematicians working with matrix theory, and anyone involved in advanced calculus or mathematical physics.

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Homework Statement


How do I show that det(I+Adt)=1+tr(A)dt +... ? Please help me :)


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The Attempt at a Solution

 
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Write I + A dt = exp(A dt). Then use the property det(exp(A)) = exp(trA). (for dt small)
 
Another way to do it using the determinant formula

det(A) = \epsilon_{i_1i_2i_3...}A_{i_11}A_{i_22}A_{i_33}...

That gives you the same result. Try it!
 

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