Algebra of Matrices — Addition, Scalar Multiplication and Multiplication

Addition and subtraction. Two matrices can be added (or subtracted) only when they are of the same order; the sum is obtained by adding corresponding elements. If $A=[a_{ij}]$ and $B=[b_{ij}]$ are both $m\times n$, then $A+B=[a_{ij}+b_{ij}]$. For instance, $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}+\begin{pmatrix} 0 & -1 \\ 2 & 5 \end{pmatrix}=\begin{pmatrix} 1 & 1 \\ 5 & 9 \end{pmatrix}.$ Matrix addition is commutative ($A+B=B+A$) and associative ($(A+B)+C=A+(B+C)$), and the zero matrix $O$ is the additive identity.

Scalar multiplication. To multiply a matrix by a scalar (an ordinary number) $k$, multiply every element by $k$: $kA=[k\,a_{ij}]$. So $3\begin{pmatrix} 1 & -2 \\ 0 & 4 \end{pmatrix}=\begin{pmatrix} 3 & -6 \\ 0 & 12 \end{pmatrix}.$ Combining the two operations, expressions such as $2A-3B$ are formed by scaling first and then adding element by element.

Matrix multiplication. The product $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$. If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ exists and is of order $m\times p$. The element in row $i$, column $j$ of $AB$ is found by the row-by-column rule: take row $i$ of $A$ and column $j$ of $B$, multiply corresponding entries, and add. In symbols, $(AB)_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{in}b_{nj}.$

For example, $\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 2 & 5 \end{pmatrix}=\begin{pmatrix} 2(1)+3(2) & 2(0)+3(5) \\ 1(1)+4(2) & 1(0)+4(5) \end{pmatrix}=\begin{pmatrix} 8 & 15 \\ 9 & 20 \end{pmatrix}.$ …