Determinants — Formula Sheet

|A| = det A

For a 1×1 matrix A = [a11], |A| = a11

For a 2×2 matrix A = [[a,b],[c,d]], |A| = ad - bc

For a 3×3 matrix A = [[a11,a12,a13],[a21,a22,a23],[a31,a32,a33]], |A| = a11(a22a33-a32a23) - a12(a21a33-a31a23) + a13(a21a32-a31a22)

Mij = determinant of matrix formed by deleting i-th row and j-th column

Cij = (-1)^(i+j) Mij

|A| = a11C11 + a12C12 + a13C13

adj A = [Cij]^T

|AB| = |A||B|

|A^n| = |A|^n

|kA| = k^n|A|

|A^T| = |A|

If a row/column is all zero, determinant = 0

A is singular if |A| = 0

A is non-singular if |A| ≠ 0

A is invertible if and only if A is non-singular

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

|A^-1| = 1/|A|

For an order n matrix with n ≥ 2, a minor is a determinant of order n - 1.

Cofactor = (-1)^(i+j) x minor