For a 2×2 matrix (begin{pmatrix}a & b\ c & dend{pmatrix}), compute (ad – bc)
For a 3×3 matrix (begin{pmatrix}a & b & c\ d & e & f\ g & h & iend{pmatrix}), compute (a(ei – fh) – b(di – fg) + c(dh – eg))
For larger matrices, use cofactor expansion along any row or column
For larger matrices, use row reduction to convert the matrix to upper triangular form
For an upper triangular matrix, multiply the diagonal entries
Track row operations:
Swapping two rows changes the sign of the determinant
Multiplying a row by a scalar multiplies the determinant by that scalar
Adding a multiple of one row to another row does not change the determinant
For matrices with a zero row or zero column, the determinant is 0
For matrices with two equal rows or two equal columns, the determinant is 0
For singular matrices, the determinant is 0
For invertible matrices, the determinant is nonzero