Last time, we discussed [here] Jacobi’s formula expresses the differential of the determinant of a matrix A in terms of the adjugate of A and the differential of A. The formula is
A more useful formula is the following
Let us firstly reprove the Jacobi formula. Assuming 
Therefore,
Now the fact that
will help us to conclude
Let us now consider the following second derivative
To this purpose, it is well-known that 
In other words,
This and the fact that 
where 
provided 
We also have the following useful identity
![\displaystyle\frac{{{d^2}}}{{d{t^2}}}{\text{det}}(A + tB) = {\text{det}}(A + tB)\left[ {{\text{tr}}{{\big((A + tB)}^{ - 1}}B{\big)^2} + \frac{d}{{dt}}{\text{tr}}{{\big((A + tB)}^{ - 1}}B\big)} \right].](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cfrac%7B%7B%7Bd%5E2%7D%7D%7D%7B%7Bd%7Bt%5E2%7D%7D%7D%7B%5Ctext%7Bdet%7D%7D%28A+%2B+tB%29+%3D+%7B%5Ctext%7Bdet%7D%7D%28A+%2B+tB%29%5Cleft%5B+%7B%7B%5Ctext%7Btr%7D%7D%7B%7B%5Cbig%28%28A+%2B+tB%29%7D%5E%7B+-+1%7D%7DB%7B%5Cbig%29%5E2%7D+%2B+%5Cfrac%7Bd%7D%7B%7Bdt%7D%7D%7B%5Ctext%7Btr%7D%7D%7B%7B%5Cbig%28%28A+%2B+tB%29%7D%5E%7B+-+1%7D%7DB%5Cbig%29%7D+%5Cright%5D.&bg=ffffff&fg=333333&s=0 "\displaystyle\frac{{{d^2}}}{{d{t^2}}}{\text{det}}(A + tB) = {\text{det}}(A + tB)\left[ {{\text{tr}}{{\big((A + tB)}^{ - 1}}B{\big)^2} + \frac{d}{{dt}}{\text{tr}}{{\big((A + tB)}^{ - 1}}B\big)} \right].")
