First of all, the number of rows and columns of determinant is the same, such as 3 rows and 3 columns, 4 rows and 4 columns, etc. In the title, A and B are M-order and N-order squares respectively, and the elements in O are all 0. I don't know if you have a test block matrix, in which A, B, O and * are all small matrices, and * is just a symbol. It is a small matrix with m rows and n columns, and its elements can be any number or all zeros. As long as all the elements in the zero matrix o are guaranteed to be zero, it is true for any conclusion.

In mathematics, Laplace expansion (or Laplace formula) is the expansion of determinant. The determinant of an n×n matrix B is expanded by Laplace, that is, it is expressed as the sum of (n- 1)×(n- 1) cofactors of a row (or a column) of the matrix B.