As a very simplified expression, an n × n complex matrix A, 1 ≤ k ≤ n, the kth compound of A is defined as the product of two same vertical matrices having two members of n and k, complex matrix Ck(A) are defined as:

Ck(A)α,β = det A[α|β]

The kth additive compound of A is usually defined as:

∆k(A) = d/dt|t=0 x Ck(I + tA)

In your question you are questioning about the Third Additive Compound Matrix , therefore for this case you apply k = 3.

Since the subject is a little complicated for application and to see portieres of it I recommend you to study some papers and some other reference books that you may search and find them.

I hope it is helpful to you and wish you the best of luck.

Sir, Kabir Sadeghi, would you please recommend me some reference books? I don't know what type of books I need to search for it.

You search the following name that have worked and written about the subject:

-Rollett and Raabe,

- Kugler and Turk,

- Raabe,

-Wang and Liu,

- Holm.

and the following references as an example:

- R. Bhatia, Matrix Analysis, Springer, New York, 1997.

- M. Fiedler, Additive compound matrics and an inequality for eigenvalues of symmetric stochastic matrices, Czechosklvak Math. J. 24 (1974) 392-402

Good luck.

Thank you so much sir, Kabir Sadeghi for you comment.

Sir, all these are keywords or authors names?

-Rollett and Raabe,

- Kugler and Turk,

- Raabe,

-Wang and Liu,

- Holm.

You are welcome.

Yes. They are.

Thank you so much Professor Kabir Sadeghi

You are most welcome and good luck..