n’th derivative of a determinant

Let $A=(a_{i,j})$ be a $d\times d$ matrix whose entries are real functions of $t$. Then,

 $\begin{split}\displaystyle\frac{d^{n}}{dt^{n}}\det(A)&\displaystyle=\sum_{n_{1% }+\cdots+n_{d}=n}{n\choose n_{1},n_{2},...,n_{d}}\sum_{\pi\in S_{d}}% \operatorname{\mathrm{sgn}}(\pi)\prod_{i=1}^{d}\frac{d^{n_{i}}}{dt^{n_{i}}}a_{% i,\pi(i)}\\ \\ &\displaystyle=\sum_{n_{1}+\cdots+n_{d}=n}{n\choose n_{1},n_{2},...,n_{d}}\det% \begin{pmatrix}\frac{d^{n_{1}}}{dt^{n_{1}}}a_{1,1}&\frac{d^{n_{1}}}{dt^{n_{1}}% }a_{1,2}&\cdots&\frac{d^{n_{1}}}{dt^{n_{1}}}a_{1,d}\cr\vdots&\vdots&&\vdots\cr% \frac{d^{n_{d}}}{dt^{n_{d}}}a_{d,1}&\frac{d^{n_{d}}}{dt^{n_{d}}}a_{d,2}&\cdots% &\frac{d^{n_{d}}}{dt^{n_{d}}}a_{d,d}\end{pmatrix}\end{split}$

where ${n\choose n_{1},n_{2},...,n_{r}}$ is the multinomial coefficient, $S_{d}$ is the symmetric group of permutations and $\operatorname{\mathrm{sgn}}(\pi)$ is the sign of a permutation $\pi$.

Title n’th derivative of a determinant NthDerivativeOfADeterminant 2013-03-22 14:30:25 2013-03-22 14:30:25 GeraW (6138) GeraW (6138) 5 GeraW (6138) Result msc 15A15 GeneralizedLeibnizRule MultinomialTheorem DerivativeOfMatrix