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Matrix theorems

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

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

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

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

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

5] ( P1P2 ... Pn )T = PnT ... P2TP1T for any matrices P1, P2, ... ,Pn 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] (AT)-1 = (A-1)T for a square, nonsingular matrix A

10] (ATB)T = BTA

11] AT = A-1 if A is orthogonal

12] AAT = (AAT)T and ATA = (ATA)T for any matrix A

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

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

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