Understanding Determinants: Calculation and Function

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What does a determinant do exactly (what is it)? And what is the general method of calculating it?
 
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The determinant is mainly used to determine whether or not a square matrix is invertible. There are other applications as well. (ie. Cramer's rule for determining solutions to a system of linear equations)

The method used is generally the http://tutorial.math.lamar.edu/Classes/LinAlg/MethodOfCofactors.aspx" (note link)
 
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hmm...I see...I've practically forgotten the concept of matrices, but thanks. If possible, do you know of a site that simply goes through the fundamentals of matrices? And (this question is slightly random)...what is a lattice point?
 
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Thanks for the help.
 

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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