For a 1×1 matrix ([a]), determinant = (a)
For a 2×2 matrix (begin{bmatrix} a & b \ c & d end{bmatrix}), determinant = (ad – bc)
For a 3×3 matrix, use expansion by minors or Sarrus’ rule
For an (n times n) matrix, expand along any row or column using cofactors
Choose an element (a_{ij})
Find its minor by deleting row (i) and column (j)
Multiply the minor by ((-1)^{i+j}) to get the cofactor
Multiply the element by its cofactor
Add the results for all elements in the chosen row or column
Use row operations to simplify the matrix before finding the determinant
Swapping two rows changes the sign of the determinant
Multiplying a row by a constant multiplies the determinant by the same constant
Adding a multiple of one row to another row does not change the determinant
For an upper or lower triangular matrix, determinant = product of diagonal entries
If a matrix has a row or column of all zeros, determinant = 0
If two rows or columns are identical, determinant = 0
If one row or column is a multiple of another, determinant = 0