Basic matrix operations

BASIC FACTS ABOUT MATRICES

[1] A + B = B + A

[2] x(A + B) = xA + xB,           where x is any number

[3] (x+y)A = xA + yB

[4] AB does not always equal BA

[5] A(BC) = (AB)C

[6] A(BC) does not always equal (AC)B (for example, consider A = I)

[7] AA^-1 = I, the Identity matrix

[8] (A^T)^T = A

[9] (A + B)^T  = A^T + B^T

[10] (xA)^T = x A^T

[11] (AB)^-1 = B^-1 A^-1

[12] (AB)^T = B^T A^T

[13] (A^-1)^T = (A^T)^-1