How to Expand a Direct Determinant in Homework?

[Grand]

Homework Statement

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

Homework Equations

The Attempt at a Solution

 

[praharmitra]

Write I + A dt = exp(A dt). Then use the property det(exp(A)) = exp(trA). (for dt small)

 

[praharmitra]

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!