Matrix theorems

A, B, C, etc. are matrices and a, b, c etc. are scalars. A^T is A transpose. A^-1 is A inverse.

 1] (AB)^-1 = B^-1A^-1 for any square, nonsingular matrices A and B

 2] ( P₁P₂ ... P_n )^-1 = P_n^-1 ... P₂^-1P₁^-1 for square, nonsingular matrices P₁, P₂, ... ,P_n

 3] (A + B)^T = A^T + B^T for any mxn matrices A and B

 4] (AB)^T = B^TA^T for any matrices A and B conformable for multiplication 

 5] ( P₁P₂ ... P_n )^T = P_n^T ... P₂^TP₁^T for any matrices P₁, P₂, ... ,P_n conformable for multiplication

 6] |AB| = |A||B| determinant of product is product of determinants for square, nonsingular matrices A and B 

 7] A^-n = (A^-1)ⁿ = (Aⁿ)^-1 for a square, nonsingular matrix A and any integer n

 8] (cA)^-1 = c^-1A^-1 where c is a scalar and A is square and nonsingular

 9] (A^T)^-1 = (A^-1)^T for a square, nonsingular matrix A

10] (A^TB)^T = B^TA

11] A^T = A^-1 if A is orthogonal

12] AA^T = (AA^T)^T and A^TA = (A^TA)^T for any matrix A

13] A + A^T = (A+A^T)^T for any square matrix A

14] If the m-square matrix A is symmetric and if P is of order mxn then B = P^TAP is symmetric.

15] If the m-square matrix A is skew-symmetric and if P is of order mxn then B = P^TAP is skew- symmetric.

16] If A and B are n-square symmetric matrices then AB is symmetric if and only if A and B commute