For a 1×1 matrix ([a]), the determinant is (a)
For a 2×2 matrix (begin{pmatrix} a & b \ c & d end{pmatrix}), compute (ad – bc)
For a 3×3 matrix, use expansion by minors or Sarrus’ rule
For an (ntimes n) matrix, expand along any row or column using cofactors
For each entry (a_{ij}), compute the minor by deleting row (i) and column (j)
Multiply each minor by ((-1)^{i+j}) to get the cofactor
Sum the products of the row or column entries with their cofactors
Use row operations to simplify the matrix before computing
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 triangular or diagonal matrices, 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 equal, the determinant is 0
If one row or column is a multiple of another, the determinant is 0
For large matrices, use Gaussian elimination to reduce to triangular form and multiply the diagonal entries, adjusting for row swaps and scaling