3×3 Determinants by Expansion

The same method works for determinants of any size. Consider the 4×4 matrix:

Choose a row or column, typically with as many zeros as possible to save multiplications.

Form terms made of three parts:

1. the entries from the row or column

2. the signs from the row or column; they form a checkerboard pattern:

3. the minors; these are the determinants of the matrix with the row and column of the entry taken out; here dots are used to show those. For example, here are the minors for the first row:

, , ,

Here is the determinant of the matrix by expanding along the first row:

- + -

The product of a sign and a minor is called a cofactor.

Even when there are many zero entries, row reduction is more systematic, simpler, and less prone to error. Row reduction on a determinant uses the three elementary row operations. If you factor a number from a row, it multiplies the determinant. If you switch rows, the sign changes. And you can add or subtract a multiple of one row from another. When the matrix is upper triangular, multiply the diagonal entries and any terms factored out earlier to compute the determinant.