How is the determinant of a 3x3 matrix calculated?

The determinant of a 3x3 matrix is calculated using the specific formula where each element of the first row is multiplied by the determinant of the 2x2 matrix that remains after removing the corresponding row and column. The formula applied here breaks down as follows:

For a matrix represented as:

[ A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} ]

The determinant is found using:

[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) ]

This formula involves the products of the elements of the first row and the corresponding sub-determinants of the 2x2 matrices formed by omitting the row and column of each element. The alternating signs in the formula account for the matrix's inherent properties, ensuring that the determinant represents the signed volume of the parallelepiped defined by the vectors in the matrix.

Thus, the correct choice accurately reflects the determinant's calculation process for a 3x3 matrix, which is distinct from simply taking the sum of the diagonal elements, performing row operations, or multiplying all entries together,