Matrices of the same dimensions can be added entry by entry. If $A$ and $B$ are both $m\times n$, then

$$(A+B){ij}=A{ij}+B_{ij}.$$

For example,

$$\begin{pmatrix}1&2\3&4\end{pmatrix} + \begin{pmatrix}5&-1\0&2\end{pmatrix}

\begin{pmatrix}6&1\3&6\end{pmatrix}.$$

Matrices with different dimensions cannot be added because their entries do not line up.

Scalar multiplication

Multiplying a matrix by a scalar multiplies every entry:

$$cA=(cA_{ij}).$$

For example,

$$2\begin{pmatrix}1&-3\4&0\end{pmatrix}

\begin{pmatrix}2&-6\8&0\end{pmatrix}.$$

Linear structure

These operations follow the same patterns as vector addition and scalar multiplication. For example,

$$A+B=B+A,$$

$$c(A+B)=cA+cB.$$

This shared linear structure is one reason matrices and vectors fit naturally into the same theory.