Properties of Matrix Addition, Scalar Multiplication and Product of Matrices

Let $A,B,C$ be matrices of orders that make the stated operation possible, and let $a,b$ be scalars.

Properties of addition and scalar multiplication:

1. $A+B$ has the same order as $A$ and $B$.
2. $A+B=B+A$ (commutative).
3. $(A+B)+C=A+(B+C)$ (associative).
4. $A+O=O+A=A$ ($O$ is the additive identity).
5. $A+(-A)=O=(-A)+A$ ($-A$ is the additive inverse of $A$).
6. $(a+b)A=aA+bA$ and $a(A+B)=aA+aB$.
7. $a(bA)=(ab)A$, $1A=A$, $0A=O$.

Properties of multiplication (assuming the stated orders):

• If $A,B,C$ are $m\times n$, $n\times p$, $p\times q$ respectively, then $A(BC)=(AB)C$ (associative), a matrix of order $m\times q$.
• If $A,B,C$ are $m\times n$, $n\times p$, $n\times p$ respectively, $A(B+C)=AB+AC$ (left distributive).
• If $A,B,C$ are $m\times n$, $m\times n$, $n\times p$ respectively, $(A+B)C=AC+BC$ (right distributive).
• For a scalar $\alpha$ and conformable $A$ ($m\times n$), $B$ ($n\times p$): $\alpha(AB)=(\alpha A)B=A(\alpha B)$.
• $AI=IA=A$, where $I$ is the unit matrix (the multiplicative identity). …