For a 2×2 matrix (begin{pmatrix} a & b \ c & d end{pmatrix}), use (det = ad – bc)
For a 3×3 matrix, use cofactor expansion or the rule of Sarrus
For an (n times n) matrix, use cofactor expansion along any row or column
For large matrices, use row reduction to convert the matrix to upper triangular form
If using row reduction, track how row operations change the determinant
Swapping two rows changes the sign of the determinant
Multiplying a row by a scalar multiplies the determinant by the same scalar
Adding a multiple of one row to another row does not change the determinant
For an upper triangular or lower triangular matrix, multiply the diagonal entries
If a matrix has a row or column of all zeros, the determinant is 0
If two rows or two columns are identical, the determinant is 0
If one row or column is a multiple of another, the determinant is 0
If the matrix is singular, the determinant is 0
Use software or a calculator for very large matrices when allowed