(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

PROVE:

If A(t) is nxn with elements which are differentiable functions of t

Then:

[itex]\frac{d}{dt}[/itex](det(A))=[itex]\sum[/itex]det(A_{i}(t))

where A_{i}(t) is found by differentiating the i^{th}row only.

2. Relevant equations

I know I should prove this by induction on n

3. The attempt at a solution

Consider the matrix A_{1}being a 1x1 matrix

So n=1

the derivative of the determinant is the same as the derivative of that one row, therefore the theorem holds for n=1

assume the proof will hold true for n=k call this matrix A_{k}

now prove the theorem holds true for n=k+1

[itex]\frac{d}{dt}[/itex](det(A_{k+1})) =[itex]\frac{d}{dt}[/itex](det(A_{1})+[itex]\frac{d}{dt}[/itex](det(A_{k}))

AND

[itex]\sum[/itex](det(A_{k+1}) = [itex]\sum[/itex]det(A_{1})+[itex]\sum[/itex]det(A_{k})

Is this it?

have I proved it?

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# Prove the derivative of the determinant equals the sum of the derivatives of ea/row

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