Let det(A) and det(B) denote the determinants of the matrices A and B,…

Key idea: swapping two rows of a matrix multiplies its determinant by −1.

How B is formed from A:

• A's first row is [1, 2, 3, 4], A's third row is [3, 4, 1, 2].

• Matrix B has these two rows swapped compared with A (its rows are in the order: third row of A, second row of A, first row of A, fourth row of A).

Conclusion:

• Because B is obtained by a single swap of two rows of A, det(B) = −det(A).

Optional numeric check:

• A is a circulant matrix with eigenvalues 10, −2−2i, −2, −2+2i (for 4th roots of unity).

• The determinant is the product of eigenvalues: det(A) = 10 · (−2−2i) · (−2) · (−2+2i) = −160, so det(B) = 160.