How to Expand a Direct Determinant in Homework?

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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!
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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