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Good morning everyone,

I would like to know how to calculate:

$\frac{d}{dt}\det \big(A_1(t), A_2(t), \ldots, A_n (t) \big)$

Help me please.

Thank you

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The formula is $$d(\det(m))=\det(m)Tr(m^{-1}dm)$$ where $dm$ is the matrix with $dm_{ij}$ in the entires. The derivation is based on Cramer’s rule, that $m^{-1}=\frac{Adj(m)}{\det(m)}$. It is useful in old-fashioned differential geometry involving principal bundles.

I noticed Terence Tao posted a nice blog entry on it. So I probably do not need to explain more at here.

Think I can provide a proof for Matias’ formula.

So, let

$$

A(t) = \mathrm{det}\left( A_1(t), \dots , A_n(t) \right) \ .

$$

By definition,

$$

\frac{dA(t)}{dt} = \mathrm{lim}_{h\rightarrow 0} \frac{A(t+h) – A(t)}{h} = \mathrm{lim}_{h\rightarrow 0} \frac{\det (A_1(t+h), \dots, A_n(t+h)) – \det(A_1(t), \dots , A_n(t))}{h}

$$

Now, we subtract and add

$$

\det(A_1(t), A_2(t+h), \dots , A_n(t+h))

$$

obtaining:

$$

\frac{dA(t)}{dt} = \mathrm{lim}_{h\rightarrow 0} \frac{\det (A_1(t+h), A_2(t+h),\dots, A_n(t+h)) – \det(A_1(t), A_2(t+h), \dots , A_n(t+h))}{h} +

\mathrm{lim}_{h\rightarrow 0}\frac{

\det(A_1(t), A_2(t+h), \dots , A_n(t+h))-\det(A_1(t), \dots , A_n(t))}{h}

$$

Now we focus on the first addend, which is

$$

\det \left( \mathrm{lim}_{h\rightarrow 0} \frac{A_1(t+h) – A_1(t)}{h}, \mathrm{lim}_{h\rightarrow 0} A_2(t+h), \dots,\mathrm{lim}_{h\rightarrow 0} A_n(t+h) \right)

$$

That is,

$$

\det (A_1′(t), A_2(t), \dots , A_n(t)) \ .

$$

Now, let’s go for the second addend to which we substract and add

$$

\det(A_1(t), A_2(t), A_3(t+h), \dots , A_n(t+h)) \ .

$$

From which we will obtain the term

$$

\det (A_1(t), A’_2(t), A_3(t), \dots , A_n(t)) \ .

$$

Keep on doing analogous operations till you get

$$

\det (A_1(t), A_2(t), \dots , A_{n-1}(t), A_n'(t)) \ .

$$

Like product rule:

$$\dfrac{d}{dt}\det(A_1(t),A_2(t),…,A_n(t))=\det(A_1^{‘}(t),A_2(t),A_n(t))+\det(A_1(t),A_2^{‘}(t),…,A_n(t))+…+\det(A_1(t),A_2(t),…,A_n^{‘}(t)) $$

In the previous answers it was not explicitly said that there is also the Jacobi’s formula to compute the derivative of the determinant of a matrix.

You can find it here well explained: JACOBI’S FORMULA.

And it basically states that:

Where the adj(A) is the adjoint matrix of A. How to compute the adjugate matrix is explained here: ADJUGATE MATRIX.

I hope it will help someone.

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