Determinant Properties:

• det($A^T$) = det(A)
• det(AB) = det(A) * det(B)
• det(kA) = $k^n$ * det(A) for n x n matrix
• det(A^(-1)) = 1/det(A)
• Swap rows: sign changes
• Add multiple of row to another: no change
• Triangular matrix: det = product of diagonal

Adjoint Identities:

• adj(A) = (cofactor matrix)^T
• A * adj(A) = det(A) * I
• det(adj(A)) = (det(A))^(n-1)
• adj(adj(A)) = (det(A))^(n-2) * A
• adj(AB) = adj(B) * adj(A)
• adj(kA) = k^(n-1) * adj(A)

Inverse:

• A^(-1) = adj(A) / det(A)
• (AB)^(-1) = B^(-1) * A^(-1)
• ($A^T$)^(-1) = (A^(-1))^T

Trace:

• tr(A + B) = tr(A) + tr(B)
• tr(AB) = tr(BA)
• For 2x2: characteristic eq is $lambda^2$ - (tr A)lambda + det(A) = 0

Cayley-Hamilton (2x2):

• $A^2$ - (tr A)A + (det A)I = O
• A^(-1) = [(tr A)I - A] / det(A)

Special Determinants:

• Vandermonde: |1 a $a^2$; 1 b $b^2$; 1 c $c^2$| = (a-b)(b-c)(c-a)
• Skew-symmetric (odd order): det = 0