Matrix Algebra

For all matrices $A_{m\times n}$, $B_{m\times n}$ and scalars c:

• c(A+B)=cA+cB
• A+B=B+A
• (A^T)^T = A
• A-A=0
• (A+B)^T = A^T+B^T

In addition, for all matrices $C_{p\times m}$:

• (AB)^T = B^T A^T
• C(A+B) = CA + CB

However, $AB \neq BA$ in general (even if both are square):

$$\left[\begin{array} {cc} 1 & 2 \ 3 & 4 \end{array} \right]... ...left[\begin{array} {cc} 10 & 4 \ 24 & 10 \end{array} \right],$$

but

$$\left[\begin{array} {cc} 4 & 2 \ 3 & 1 \end{array} \right]... ...left[\begin{array} {cc} 10 & 16 \ 6 & 10 \end{array} \right].$$