04 - Matrix Addition and Scalar Multiplication

Matrix Addition

Let $A=(a_{ij})$ and $B=(b_{ij})$ be $m\times n$ matrices

$C=A+B=(c_{ij})$ such that

$$c_{ij}=a_{ij}+b_{ij}$$

for $1\le i\le m$ and $1\le j\le n$

Requires matrices of the same size

Property of Matrix Addition (for same-size matrices $M,N,L$)

• Commutative: $M+N=N+M$
• Associative: $L+(M+N)=(L+M)+N$

Matrix Addition with zero matrix $0_{mn}$ and $m\times n$ matrix $A$

$$A+0_{mn}=A=0_{mn}+A$$

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Scalar Multiplication

Let $A=(a_{ij})$ be a $m\times n$ matrix and $k\in \mathbb{R}$

$B=kA=(b_{ij})$ such that

$$b_{ij}=k{a}_{ij}$$

for $1\le i\le m$ and $1\le j\le n$

Distributivity of Scalar Multiplication over Matrix Addition

$$k(A+B)=kA+kB$$

where $A,B$ are $m\times n$ matrices and $k\in \mathbb{R}$

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General Operation Identities for Matrix Addition and Scalar Multiplication

Let $A,B,C$ be $m\times n$ matrices and $\lambda ,\mu \in \mathbb{R}$

$$\begin{array}{rlrlrlrlrlrl}& A+0_{mn}& & =A;& & A+B& & =B+A;& & 0A& & =0_{mn}\\ & A+(-A)& & =0_{mn};& & (A+B)+C& & =A+(B+C);& & 1A& & =A\\ & (\lambda +\mu )A& & =\lambda A+\mu A;& & \lambda (A+B)& & =\lambda A+\lambda B;& & \lambda (\mu A)& & =(\lambda \mu )A\end{array}$$

These show that $M_{mn}$ is a real vector space