Which is a formula for calculating the derivative of a determinant with respect to one of its elements? How do I prove this?

I know the result has something to do with the inverse, but I can't just quite grasp the proof and can't find it anywhere

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- Aug 27th 2009, 10:00 PMaltave86Derivative of a determinant
Which is a formula for calculating the derivative of a determinant with respect to one of its elements? How do I prove this?

I know the result has something to do with the inverse, but I can't just quite grasp the proof and can't find it anywhere - Aug 27th 2009, 11:47 PMred_dog
Let

Then

- Aug 28th 2009, 03:56 AMhalbard
If is an invertible matrix which depends on a real parameter , and if exists, then

, where is the trace of .

I have discovered a truly marvellous proof of the above statement, which unfortunately my brain is too small to contain.