How to compute the determinant of a matrix:

The determinant is the sum of product terms made up of elements from the matrix.

Each product term consists of n elements from the matrix.

Each product term includes one element from each row and one element from each column.

The number of product terms is equal to n! (where n! refers to n factorial).

By convention, the elements of each product term are arranged in ascending order of the left-hand (or row-designating) subscript.

To find the sign of each product term, we count the number of inversions needed to put the right-hand (or column-designating) subscripts in numerical order. If the number of inversions is even, the sign is positive; if odd, the sign is negative.