Determinants

|A| = det A = Δ

For a 1×1 matrix A = [a11], |A| = a11

For A = [a11 a12; a21 a22], |A| = a11a22 - a12a21

For A = [a11 a12 a13; a21 a22 a23; a31 a32 a33], |A| = a11a22a33 - a11a32a23 - a12a21a33 + a12a31a23 + a13a21a32 - a13a31a22

Expand along a row or column using minors and cofactors

If A = kB, then |A| = k^n|B|

|A| = sum of the products of elements of any row or column with their corresponding cofactors

Δ = 1/2 |x1 y1 1; x2 y2 1; x3 y3 1|

adj A = transpose of [Aij]

|adj(A)| = |A|^(n-1)

A^{-1} = (1/|A|) adj A

AB = BA = I

If AB = I, then |A| ≠ 0